How Does Pressure Change with Volume in an Ideal Gas Expansion Scenario?

[mmwave]

An unknown number of Helium atoms are in a 1 liter container at an unknown temperature. The gas is made to expand such that it's final volume is 4 liters and pressure rises 'in direct proportion to its volume'.

1. does 'in direct proportion to it volume' mean Pf = Pi * Vf/Vi or should there be a proportionality constant a?

2. I calculated the work done on the gas by using P(v) = a*V and using W = -[inte] aVdV = -a/2 * (Vf^2 - Vi^2) Comments?

3. I am having more trouble getting change in internal energy dU.

I tried PV = NkT so PfVf - PiVi = Nk( Tf - Ti) and
dU = 3Nk (Tf - Ti)/2 so that

dU = 3/2 * (PfVf-PiVi) = 3/2 * (4*4*aVi - Vi)

Is this even close?

4. If I can get 2 and 3 I can calculate Q myself!

 

[HallsofIvy]

"Pressure is in direct proportion to Volume" MEANS that P is some constant times V: P= aV where a is the constant of proportionality.

HOWEVER, if you write P1= aV1 and P2= aV2 for two different sets of pressure and volume, and then divide one by the other: P1/P2= V1/V2 (which is the equation you are using) does not involve a constant of proportionality because they cancel.

2) looks correct to me.


In 3) You use PV= NRT which says that for CONSTANT temperature, P is INVERSELY proportional to V. But the problem says P is directly proportional to V! Well, of course, that says that the temperature does NOT stay constant! You can use PV= NRT together with P= aV to find the change in temperature just as you did.

 

[M98Ranger]

1. "In direct proportion to its volume" means that the pressure and volume are directly proportional to each other, without the need for a proportionality constant. Therefore, the equation would be Pf = Pi * Vf/Vi.

2. Your calculation for work done on the gas seems correct. The negative sign indicates that work is being done on the gas, which makes sense since the gas is expanding and increasing in volume.

3. Your approach for calculating the change in internal energy is not entirely correct. The equation PV = NkT is for an ideal gas, but in this case, we do not know the temperature or number of particles. Instead, we can use the ideal gas law to relate the initial and final pressures and volumes: PiVi = NkTi and PfVf = NkTf. We can then subtract these equations to get PfVf - PiVi = Nk(Tf - Ti). We can then use the specific heat capacity of an ideal gas at constant volume (Cv = 3/2 Nk) to calculate the change in internal energy: dU = Cv(Tf - Ti). Therefore, dU = 3/2 * (PfVf - PiVi) = 3/2 * (4*4*aVi - Vi) = 9/2 * (a * Vi^2).

4. Now that you have the correct equation for change in internal energy, you can use it to calculate the heat transfer (Q) using the first law of thermodynamics: Q = dU + W. Plug in the values for dU and W that you have calculated to get the final answer for Q.