Homework Question 9.13

[Paula Dowdell 1F]

"During the test of an internal combustion engine, 3.00 L of nitrogen gas at 18.5 C was compressed suddenly (and irreversibly) to 0.500 L by driving in a piston. In the process, the temperature of the gas increased to 28.1 C. Assume ideal behavior. What is the change in entropy of the gas?"

How come when determining the change in volume Cv (5/2 R) is used but when determining the change in temperature just R is used?

Thanks

[Isabella Sanzi 2E]

So according to what I see in the solution manual, all you have to do is use the formulas deltaS = nRln(V2/V1) and deltaS = nRln(T2/T1), so you don't have to worry about a constant for C at all. Just plug in what is given to solve for the entropy for the whole system based on each step of the reaction taking place. I hope this helps!

[Lisa Tang 1C]

I think you actually mean R is used when volume changes and C_v when temperature changes. The equations in which these values are plugged in are both derived from others. On pages 321 and 323, the derivations are explained for the equation for entropy change of a system with temperature change and one of volume change, respectively.
For a change in temperature, the energy supplied as heat is related to the increase in temperature by the heat capacity, C.
For a change in volume, the equation relates to w_rev=-nRT*ln, which is equal to heat released (but with the opposite sign). Using the equation S=q_rev/T=(-w_rev)/T, you can substitute nRT*ln in for -w_rev and end up with nR*ln.

[Paula Dowdell 1F]

But how do you know that "for a change in temperature, the energy supplied as heat is related to the increase in temperature by the heat capacity, C". Is that always true?

[Lisa Tang 1C]

But how do you know that "for a change in temperature, the energy supplied as heat is related to the increase in temperature by the heat capacity, C". Is that always true?

The definition of heat capacity is the ratio of the heat supplied to the rise of temperature produced: C= q/T. So, this should always be true.