Chemistry To determine variables in adiabatic reversible process do we need to look up heat capacity?

[zenterix]

Homework Statement

I have a quick question about a relatively simple problem.

Suppose we have 1 mole of a specified gas (say, helium) and it undergoes a reversible adiabatic process.

We are given initial volume $V_1$ and final volume $V_2$ and initial temperature $T_1$.

Relevant Equations

If we want to calculate $T_2$ and $P_2$, is is mandatory that we determine what $\gamma=\frac{C_P}{C_V}$ is?

This problem does not specify if helium is to be treated as an ideal gas. I am assuming it is an ideal gas.

We can determine $P_1$ from the ideal gas law

$$P_1=\frac{nRT_1}{V_1}\tag{1}$$

Since the process is adiabatic we have $\delta q=0$ and

$\dU=C_V dT=-PdV=-\frac{nRT}{V}dV\tag{2}$

After integrating and some algebra we reach

$$\frac{T_2}{T_1}=\left (\frac{V_1}{V_2}\right )^{\gamma-1}\tag{3}$$

From this we can sub in for temperature using the ideal gas law to reach

$$\left (\frac{V_1}{V_2}\right )^\gamma=\frac{P_2}{P_1}\tag{4}$$

$$PV^\gamma=\text{constant}=k\tag{5}$$

We don't know what $k$ because we don't know what $\gamma is$.

If we knew gamma we would have $k$ from $P_1V_1^\gamma$ and we could then compute $P_2$.

We could also then compute $T_2$ from (3).

To find $\gamma$ we can do

$$\gamma=\frac{C_P}{C_V}=\frac{C_P}{C_P-nR}$$

and so we just need to look up $C_P$ for helium at $T_1$.

In the integration to reach (3) we already assumed that $C_V$ is constant which means $C_P=C_V+nR$ must be constant.

This is how I solved the problem.

I just want to make sure there isn't another way. Do we need to look up heat capacity to solve this problem in this way or is there another way?

 

[Chestermiller]

Cp and Cv are heat capacities per mole, so nCvdT=~PdV and Cp-Cv=R. Also, He is monoatomic, so Cv=1.5R