A very puzzling problem

[BudsPro+]

Homework Statement

Material X,which is at a higher temperature, and Mercury are brought into contact for some time to allow heat transfer to take place.


C_X : Heat capacity of Material X
θ_X : Temperature of Material X (At first)
C_M : Heat capacity of Mercury
θ_M : Temperature of Mercury (At first)

θ_X is greater than θ_M. [ θ_X > θ_M ]
So heat energy is transferred from Material X to Mercury.
θ_N : Final temperature of both Material X and Mercury after heat transfer is complete. [ At thermal equilibrium. ]

Assumption: There is no heat loss to surroundings.

Homework Equations

Heat gained by mercury = Heat lost by Material X
δQ_M = - δQ_X
C_Mδθ_M = - C_Xδθ_X
C_M(θ_N – θ_M) = - C_X(θ_N – θ_X)
θ_NC_M – θ_MC_M = θ_XC_X - θ_NC_X
θ_NC_M + θ_NC_X = θ_XC_X + θ_MC_M

θ_N = [ θ_XC_X + θ_MC_M ] / [ C_X + C_M ]

The Attempt at a Solution

Therefore if we take X to be of different materials or even of different masses, the final temperature of the mercury , θ_N , should vary because the heat capacity of X , C_X , changes. However this is not the case as if, for example, we place 2 mercury-in-glass thermometers in Ethanol at 50^oC and Water at 50^oC, both thermometers show a reading of 50^oC.
Note: The Ethanol and the Water have different heat capacities.

I would like to see what you think about this problem. I really cannot understand, according to the above formula, why the thermometers show the same reading for the Ethanol and the Water. Any 2 other substances could have been taken but Ethanol and Water are simpler. Could someone please give me a good explanation about this?

 

[Doc Al]

Therefore if we take X to be of different materials or even of different masses, the final temperature of the mercury , θ_N , should vary because the heat capacity of X , C_X , changes.

Yes, the final temperature will depend on the mass and heat capacity of the substance.

However this is not the case as if, for example, we place 2 mercury-in-glass thermometers in Ethanol at 50^oC and Water at 50^oC, both thermometers show a reading of 50^oC.

I don't understand the point you are trying to make. Can you restate it?

Regardless of the heat capacities of the two substances, sooner or later they will reach thermal equilibrium which means they have the same temperature. That final will be different for different substances, of course.

 

[BudsPro+]

what I mean is that if for example, we take mercury(20 degrees celcius, in fact room temperature) & in contact with water(50 degreees celcius) and on the other side we take the same mercury at room temperature but in contact with Ethanol at 50 degrees celcius, the final temperature reached in the 2 cases will be different. So why do the 2 thermometers(which contain mercury at 20 degrees celsius), placed in separate beakers of Water at 50 degrees and Ethanol at 50 degrees, show the same reading of 50 degrees celcius. The reading should have been different. Don't you think so?

 

[Doc Al]

So why do the 2 thermometers(which contain mercury at 20 degrees celsius), placed in separate beakers of Water at 50 degrees and Ethanol at 50 degrees, show the same reading of 50 degrees celcius. The reading should have been different. Don't you think so?

Ah... now I see what you're saying. Technically, you are correct. But assume that the amount of mercury is small enough that the amount of heat needed to increase the thermometer's temperature is too small to make a measurable difference in the final temperature of the substance in the beaker.

 

[BudsPro+]

Yeahh thx. I have tried it and it works for small masses of mercury though there is still a little difference in the reading ( of about 0.5 degrees celcius). So would you agree if I say that the temperature reading given by a liquid-in-glass thermometer is not the true temperature of the body being measured ?

 

[Doc Al]

So would you agree if I say that the temperature reading given by a liquid-in-glass thermometer is not the true temperature of the body being measured ?

Once they have reached thermal equilibrium, the measured temperature will be the 'true' temperature of the body. But it won't necessarily be the temperature of the body before the thermometer was introduced.