What is Volume: Definition and 1000 Discussions

Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or 3D shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container; i.e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces.
Three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, and circular shapes can be easily calculated using arithmetic formulas. Volumes of complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.
The volume of a solid (whether regularly or irregularly shaped) can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas. The combined volume of two substances is usually greater than the volume of just one of the substances. However, sometimes one substance dissolves in the other and in such cases the combined volume is not additive.In differential geometry, volume is expressed by means of the volume form, and is an important global Riemannian invariant.
In thermodynamics, volume is a fundamental parameter, and is a conjugate variable to pressure.

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  1. EastWindBreaks

    How to find specific volume given temperature?

    Homework Statement Homework Equations The Attempt at a Solution I don't understand how the solution got its values, not from the tables right? because I couldn't find such value from all of the tables. Since the pressure is less than the saturated pressure at given temperature, it is in...
  2. karush

    MHB M2215b.08 Find the volume of the solid (shell meithod)

    $\tiny{M2215b.08}$ Find the volume of the solid \begin{align*}\displaystyle y&=\sin (x^2)\\ 0&\le x \le \frac{\pi}{2}\\ \end{align*} about the y-axis ok this looks like a cylindrical Shell solution So I set it up like this,,,,, hopefully $\begin{align*}\displaystyle V&=\int_0^12\pi...
  3. J

    Conversion in equal volume CSTR and PFR

    Hello I've been working on a problem that involves finding which reaction order gives a greater conversion for a CSTR over a PFR (Xcstr>Xpfr). I was able to solve the mass balance for each reactor, finding: and Allowing us to equate them. This is where I've been stuck for a while now, how...
  4. Jamie_Pi

    Calculate Final volume and temperature of piston

    Homework Statement One cylinder in the diesel engine of a truck has an initial volume of 640 cm3. Air is admitted to the cylinder at 30°C and a pressure of 1.0 atm. The piston rod then does 500 J of work to rapidly compress the air. What are its final temperature and volume? Homework Equations...
  5. D

    Finding the volume surrounded by a curve using polar coordinate

    Homework Statement I tried to answer the following questions is about the curve surface z= f (x, y) = x^2 + y^2 in the xyz space. And the three questions related to each otherA.) Find the tangent plane equation at the point (a, b, a^2+ b^2) in curved surface z . The equation of the...
  6. Leo_lee

    Calculate the volume of 1.00 mol of liquid water

    Homework Statement Calculate the volume of 1.00 mol of liquid water at 20°C (at which its density is 998 kg>m3),and compare that with the volume occupied by 1.00 mol of water at the critical point, which is 56 * 10-6 m3. Water has a molar mass of 18.0 g>mol. Full answer please! Thanks...
  7. S

    Work and Volume in Adiabatic Expansion

    Homework Statement A sample of 1.60 mol H2 (Cv = 20.5 J K-1 mol-1) at 21°C and 1.50 atm undergoes a reversible adiabatic compression until the final pressure is 4.50 atm. Calculate the final volume of the gas sample and the work associated with this process. Assume that the gas behaves ideally...
  8. S

    Volume of Double Integral: Finding the Region with Graphed Equations

    Homework Statement z=x^2+xy ,y=3x-x^2,y=x find the volume of the region Homework EquationsThe Attempt at a Solution I graphed y=3x-x^2 and y=x I am confused on which region I use to find the volume. Do I use the upper region or the lower region.
  9. yecko

    Triple integration - find volume

    Homework Statement Homework Equations in the pic The Attempt at a Solution why is there an extra "r" in the highlighted line? my attempt: ## \int_0^{2\pi}\int_0^2\int_{r^2}^41\ dy\ r\ dr\ d\theta ## = ##\int_0^{2\pi}\int_0^2\left(4-r^2\right)\ r\ dr\ d\theta## thanks
  10. K

    What is the maximum volume of a cone?

    Homework Statement Homework Equations First derivative=maxima/minima/vertical tangent/rising/falling Volume of a cone: ##~\displaystyle V=\frac{\pi}{3}r^2h## The Attempt at a Solution $$L=a^2+b^2~\rightarrow b^2=L-a^2$$ $$V=\frac{\pi}{3}a(L-a^2)$$...
  11. Muthumanimaran

    Finding the probability of 1s electron within a cubical volume

    Homework Statement How to calculate the probability of finding an 1s electron within 1 picometer cubic region located 50pm from the nucleus. Homework Equations The probability of an 1s electron within a spherical volume of radius 'a' from nucleus can be find using the expression...
  12. karush

    MHB 18 Find the volume of the solid generated by revolving the region about y-axis.

    $\textsf{Find the volume of the solid generated by revolving the region about y-axis.Given the boundares of }\\$ \begin{align*} \displaystyle y&=4x-x^2\\ y&=x\\ \end{align*} Ok, I presume since this is rotated around the y-axis that we have to rewrite the equations in terms of y $y=-(x^2-4x)$...
  13. A

    MHB A question on matrix's eigenvalue problem from Eberhard Zeidler's first volume of Nonlinear Function

    The question is posted in the following post in MSE, I'll copy it here: https://math.stackexchange.com/questions/1407780/a-question-on-matrixs-eigenvalue-problem-from-eberhard-zeidlers-first-volume-o I have a question from Eberhard Zeidler's book on Non-Linear Functional Analysis, question...
  14. K

    Maximum volume from a rectangular cardboard

    Homework Statement Homework Equations First derivative=maxima/minima/vertical tangent/rising/falling When f'(x)>0 → the function rises The Attempt at a Solution $$V=(15-2a)(8-2a)=4a^2-46a+120$$ $$V'=8a-46,~~V'=0\rightarrow a=5.75$$ But: ##~2a<8,~~V(a=4)=0## So 5.75>4 And: ##~V''=8## so it...
  15. karush

    MHB T6.1.1 Find the volume of the solid

    $\tiny{t6.1.1}$ $\text{The solid lies between planes perpendiaular to the}$ $\text{$x$-axis at $x=0$ and $x = 4$.}$ $\text{The cross-scctions perpendicular to the axis on the interval $0 \le x \le 4$}$ $\text{are squrares whose diagonals run for the parabola $\displaystyle f_a(x)=-\sqrt{x}$ to...
  16. M

    Proving the equation for the height of a cylinder

    Homework Statement Consider a sphere of radius A from which a central cylinder of radius a (where 0 < a < A ) has been removed. Write down a double or a triple integral (your choice) for the volume of this band, evaluate the integral, and show that the volume depends only upon the height of the...
  17. J

    Thermodynamics help in finding Temp, Volume and Pressure

    Hey Guys! I need some help for solving question 2 of this example as my HSC exam is in 5 days and I want to be able to solve this question if I receive it on the paper. Question: Graph: Answer for Q1 (Just the formula) : Work I have done towards this question: I have realized that from...
  18. J

    Pressure, Volume, and Temperature at a Point in a Thermodynamic Cycle

    Homework Statement Question: https://imgur.com/a/8rUw1 Graph: https://imgur.com/a/Z1oBO Homework Equations Answer for Q1 (Just the formula) : https://imgur.com/a/uesqN The Attempt at a Solution I have realized that from 1-2, there is an increase in pressure, decrease in volume and increase...
  19. M

    MHB Calculate the volume of the space D

    Hey! :o I want to draw the space $D$ that is between the paraboloid with equation $z=x^2+y^2$ and the cone with equation $z=\sqrt{x^2+y^2}$ and I want to calculate its volume. How does $D$ look like? To calculate the volume we have to calculate the integral $\iiint_D dxdydz$. I have done...
  20. K

    Why Does the Chain Rule in Thermodynamics Include a Negative Sign?

    I'm trying to understand why $$\left(\frac{\partial P}{\partial T}\right)_V = -\left(\frac{\partial P}{\partial V}\right)_T \left(\frac{\partial V}{\partial T}\right)_P$$ where does the minus sign come from?
  21. N

    How to find volume of irregular & porous material spong

    < Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown > How to find volumeof irregular and porous material like sponge?
  22. C

    I Surface Area of Volume of Revolution

    The problem is, find the surface area of the volume of revolution generated by rotating the curve y=e2x between x=0 and x=2 about the x-axis. Here's what I have so far... SA=∫y√(1+y2)dx =∫e2x√(1+4e4x)dx and from here I'm not really sure what to do. Any help would be appreciated.
  23. D

    Find v(z) given volume charge density

    Homework Statement Consider the parallel plate capacitor(no figure). The capacitor plates have a separation distance d in the z-direction. The volume charge density is given by ρ_v. Find v(z).Homework Equations ρ_v = ρ_o*sin(pi*z/2*d) The Attempt at a Solution [/B] ∫(ρ_vdz) =...
  24. M

    B Surface volume of 3-sphere with radius of 2 Planck length?

    What is the surface area ("surface volume") of a 3-sphere having a radius of 2 Planck lengths? Is the product of the Planck's constant, Einstein's proportionality constant and Planck time also equal to this volume? Does this equivalence signify anything? What does it signify?
  25. karush

    MHB *15.4.17 Volume between a cone and a sphere

    $\tiny{15.4.17}$ Find the volume of the given solid region bounded below by the cone $z=\sqrt{x^2+y^2}$ and bounded above by the sphere $x^2+y^2+z^2=128$ using triple integrals $\displaystyle\int_{a}^{b}\int_{c}^{d} \int_{e}^{f} \,dx\,dy \,dz$ not real sure where to to start with this?
  26. M

    B Can Increasing Human Body Density Enable Breaking Rocks by Hand Compression?

    Hello, I've been watching too much anime. I know that isn't the best way to ask a question scientific in nature, but visualizing bizarre situations is one of the best ways to come up with interesting questions. Anyways, in order to break a rock by compressing it with your hands, you would need...
  27. P

    Volume Flux for a Hydraulic Jump

    Homework Statement Problem is given in this image, https://gyazo.com/454370ff9549dcd7c53604ebfe5df105 Homework Equations Continuity or conservation of mass equation: \frac{\partial u}{\partial x} + \frac{\partial w}{\partial z} = 0 Where u is the horizontal velocity and w is the vertical...
  28. RJLiberator

    Volume Charge Density in Spherical and Cylindrical Coordinat

    Homework Statement Consider a ring of radius R placed on the xy-plane with its center at the origin. A total charge of Q is uniformly distributed on the ring. a) Express the volume charge density of this configuration ρ(s,Φ,z) in cylindrical coordinates. b) Express the volume charge density of...
  29. C

    Question Related To Volume Strength Of H2O2.

    1. Question Statement and details : 500 mL of 56V H2O2 is kept in an open container due to which some H2O2 is decomposed and evolves 8gm O2 simultaneously, during the process some H2O also vapourizes.Due to all these changes final volume is reduced by 20%. Find final volume strength of H2O2...
  30. C

    Volume of Gasses Obtained by Electrolysis of Salt Solution

    Hello everybody!How can I calculate the volume of gasses produced by electrolysis of an aqueous solution of NaNO3 if the mass fraction of the salt changed from 40% to 60% (before electrolysis the mass fraction of the salt was 40% and after electrolysis it changed to 60%)? Pressure and...
  31. RJLiberator

    Thermodynamics of a Constant Volume Gas Thermometer

    Homework Statement A constant volume gas thermometer contains a gas whose equation of state is (p+\frac{a}{V^2_m})(V_m-b)=RT and another, of identical construction, contains a different gas which obeys the ideal gas law, pV_m = RT. The thermometers are calibrated at the ice and steam points...
  32. D

    Does the volume of water affect efficiency of heater?

    Ok, so i did a physics experiment a few days ago and now i have to do a write up. The question is does the volume of water in the calorimeter affect the efficiency of the immersion heater? Could someone pls explain to me what happens & y? My results showed that as you increase the volume, the...
  33. karush

    MHB Compute The Volume Of The Solid

    Compute the volume of the solid bounded by the planes; $x=7, z=y-2, z=-2y-2, z=0, \mbox{ and } z=2$ Another source said the triple should be but Ii thought it should be $dx\,dy\,dz$ $$I = \displaystyle \int_0^7 \int_0^2 \int_{-\frac{z}{2} - 1}^{z+2} dy\,dz\,dx = 63$$ Hopefully this is the...
  34. W

    How do I convert meters cubed to cubic centimeters?

    This comes from the Intro Homework to my AP Physics 1 class. The teacher said that it is more important to figure out how to use the website than get the correct answers to the questions. However, I want to know how to get the correct answer, especially since I took chemistry last year and...
  35. mastermechanic

    Is the Combustion Chamber Volume for a Lamborghini Aventador Really That Small?

    I was analyzing the Lamborghini Aventador's tech. specs. and it is given Bore x Stroke 95mm x 76.4mm and compression ratio 11.8:1. Then I calculated the volume of displacement and it's 541.5 cm^3 for each cylinder. Then I know the compression ratio as $$\frac{V_1 + V_2}{V_2} = C.R$$ $$...
  36. T

    Volume of a sphere in cylindrical coordinates

    Homework Statement A sphere of radius 6 has a cylindrical hole of radius 3 drilled into it. What is the volume of the remaining solid. The Attempt at a Solution [/B] I am able to solve this using cylindrical coordinates but I'm having trouble when I try to solve it in spherical coordinates...
  37. P

    Pressure Decay from a Fixed Volume

    If an ideal gas is leaking from a fixed volume, does the temperature of the air in the fixed volume go down, or does it remain constant? We do pressure decay measurements under choked leakage flow conditions, but I wonder if our assumption of constant temperature for the gas remaining in the...
  38. Q

    I Shear Rate in a Rectangular Channel

    Say you have a rectangular channel with a width, w (m), and a height, h (m) and an infinite length. The channel itself is fixed and none of the sides can move with respect to one another. If you know the volume flow rate, V (m3/s) of fluid through the channel, how do you calculate the shear rate...
  39. M

    B Volume of a Cube: Definition & Explanation

    Suppose if we have a cube: The volume of the cube is the product of the length, width and the height. All this time, I've been looking at it as: To get the volume, multiply the area of the cross section of the cube by how many "layers" it has. To elaborate with the diagram given, one can see...
  40. Rotnort

    Does Changing Vector Components Affect Parallelepiped Volume?

    Homework Statement The volume of a parallelepiped defined by the vectors w, u, \text{ and }v, \text{ where } w=u \times v is computed using: V = w \cdot (u \times v) However, if the parallelepiped is defined by the vectors w-u, u, \text{ and }v, \text{ where } w=u \times v instead, the volume...
  41. HelloCthulhu

    What is the molality of 1M NH4OH in 70.092ml of water?

    Homework Statement What is the molality of 1M NH4OH dissolved in 70.092ml of water? What is the total mass of the solution? What is the total volume? What is the concentration percent? What is the molar concentration? 1M NH4OH = 35.046g density of water = 1g/ml density of NH4OH = 0.88 g/ml...
  42. dpeff

    I Given a fixed volume of pressurized air, calculate the flow rate

    I have a work related question. Although it seems like a relatively simple question, I am having difficulties finding the correct equation(s) to calculate the answer. Question: Given a fixed volume air tank filled with pressurized air, how does one calculate the flow rate over time through...
  43. D

    Equilibrium volume of two differential van der Waal gases

    Homework Statement Two ideal van der Waals fluids are contained in a cylinder, separated by an internal moveable piston. There is one mole of each fluid, and the two fluids have the same values of the van der Waals constants b and c; the respective values of the van der Waals constant ''a'' are...
  44. Const@ntine

    Compute the Work: Pressure analogous to Volume

    Homework Statement One mole of an ideal gas is warmed slowly, so that the pressure and volume go from (Pi, Vi) to (3Pi, 3Vi), in such a way that the pressure is analogous to the volume. a) What's the Work (W)? b) What is the correlation between the temperature and the volume during that...
  45. G

    How much does air volume change with temperature?

    If I have a closed cylinder with a fixed air volume inside and a piston on the open end I need to calculate how much the piston will move as the temperature changes through a range of 100 degrees F? Air volume vs temperature?
  46. F

    Sound Waves -- propagating, dispersing, volume effects?

    Generally sound waves are depicted as simple sine waves, where volume is related to amplitude, and there is periodic motion. Realistically sound waves aren’t as simple. I attached a picture of a dissipating sound wave. I would appreciate if you guys could answer a couple questions I have about...
  47. psuedoben

    Measure the volume of a lighter than air balloon

    hello, i am trying to calculate the volume of a balloon (which is quite large). It has been filled with helium via a valve connecting helium storage tanks to the balloon. The knowns I have are the volume of the storage tanks, the intital pressure in the tanks, and the final pressure in the tanks...
  48. M

    Expected decay time of a sound based on its relative volume?

    I am working on a synthesizer to model sounds like plucked strings (guitars) and struck membranes (drums). With such instruments, the harder you strike them, the longer the sound takes to decay to an inaudible level. What equation would allow one to predict the proportionate time decay of such...
  49. karush

    MHB S6.12.4.35 Find the volume of the parallelepiped

    $\tiny{s6.12.4.35}$ $\textrm{Find the volume of theparallelepiped with adjacent edges $PQ, PR$, and $PS$.}$ \begin{align*}\displaystyle P(2,0,-1),& \, Q(4,1,0), \, R(3.-1.1), S(2,-2,2)\\ \end{align*} ok I know the problem is basically solved with \begin{align*}\displaystyle...
  50. DaveC426913

    Troubleshooting Unexpected Volume Control on Smart TV and Soundbar

    I know this is so vague as to be unanswerable - at least no answer better than I might come up with through a bit of experimentation - but it's late and I'm lazy. After having my new Smart TV with soundbar for several months now, it has spontaneously started doing something different. The new...
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