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

AI Thread Summary
To determine variables in an adiabatic reversible process for helium, it is assumed to behave as an ideal gas. The ideal gas law is used to find initial pressure, and since the process is adiabatic, the change in internal energy relates to temperature and volume changes. The relationship between temperatures and volumes is derived, leading to the conclusion that knowing the heat capacity ratio, gamma (γ), is essential for calculating pressure and temperature changes. To find γ, the specific heat capacities (C_P and C_V) must be referenced, with helium's C_V being 1.5R. Ultimately, looking up the heat capacity is necessary to solve the problem accurately.
zenterix
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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?
 
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Cp and Cv are heat capacities per mole, so nCvdT=~PdV and Cp-Cv=R. Also, He is monoatomic, so Cv=1.5R
 
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