Adiabatic volume change of an ideal gas thought process

I am struggling with trying to solve the following problem, attempting just (a) and the first part of (b) for this exercise:

During a reversible adiabatic volume change of an ideal gas, $PV^{gamma}$ remains constant, where $gamma$ is the ration of the heat capacity at constant volume to that at constant pressure. i.e. $gamma = C_P/C_V$. One mole of an ideal gas, initially at $300^{circ}K$ and atmospheric pressure, is compressed to half it’s initial volume.

(a) Derive a general expression for the work done on the gas in terms of the starting and finishing volumes and other necessary variables. Remember that you’ll need to find an expression for $P$ as a function of $V$, and that this expression will involve $P_1$ and $V_1$.

(b) Use yor expression to calculate the amount of work done for a monatomic ideal gas and (b) a diatomic gas, in which the vibrational motion is completely frozen out.

Here’s my initial thought process:

• This is an adiabatic process, so $PV^{gamma} = K$ where $K$ is a constant.
• This gas is monatomic, so $gamma = 1 + 2/3$.
• We are not given a volume, but given a pressure and temperature, which means the volume can be deduced from the ideal gas law.
• This is an adiabatic volume change, so the work can be found given the following integral:

$$W = – int_{V_1}^{V_2} PdV$$

• Since $V_2 = 1/2 V_1$, we can replace that for $V_2$, and note $P=frac{K}{V^gamma}$. Thus we want to solve:

$$W = – Kint_{V_1}^{0.5V_1} frac{dV}{V^gamma}$$

This should reduce to the following equation:

$$W = -K left(frac{0.5V_1^{1-gamma} – V_1^{1-gamma}}{1-gamma}right)$$

With $V_1 = frac{nRT}{P}$ with $P = 101325 Pa$, $T = 300 K$, and $n = 1$ mole, $V = 0.0246 m^3$. This renders $V_2 = 0.0123 m^3$. $K = PV^{gamma}, therefore K = 210.83$.

After throwing that mess into a calculator, my final value is $1869.405 J$ roughly.

Some questions of mine:

• Why can we not use $P = frac{nRT}{V}$ as our function of $V$ for the integral instead of $P=frac{K}{V^gamma}$?

• My answer should not be negative, which should be due to $0.5 V_1 – V_1 = -0.5 V_1$, but I don’t see how that logic was wrong. What’s the problem there?

• Did I do anything laughably wrong here, and if so, where? I feel like I did something wrong.