A gas with `(c_(p))/(c_(V))=gamma` goes from an intial state `(p_(1), V_(1), T_(1))` to a final state `(p_(2), V_(2), T_(2))` through an adiabatic process. The work done by the gas is

Options

A. `(nR(T_(1)-T_(2)))/(gamma-1)`
B. `(p_(1)V_(1)-p_(2)V_(2))/(gamma-1)`
C. `(p_(1)V_(1)-p_(2)V_(2))/(gamma+1)`
D. `nRT (T_(1)-T_(2))`

Answer

To find the work done by the gas during an adiabatic process, we can follow these steps:

### Step 1: Understand the Adiabatic Process
In an adiabatic process, there is no heat exchange with the surroundings. The relationship between pressure (P), volume (V), and temperature (T) for an ideal gas can be described using the adiabatic condition:
\[ PV^\gamma = \text{constant} \]
where \(\gamma = \frac{C_p}{C_v}\).

### Step 2: Express Work Done in Terms of Volume
The work done \(W\) by the gas during an adiabatic process can be expressed as:
\[ W = \int_{V_1}^{V_2} P \, dV \]

### Step 3: Substitute Pressure Using the Adiabatic Condition
From the adiabatic condition, we have:
\[ P = \frac{K}{V^\gamma} \]
where \(K\) is a constant. Therefore, we can rewrite the work done as:
\[ W = \int_{V_1}^{V_2} \frac{K}{V^\gamma} \, dV \]

### Step 4: Perform the Integration
Now we can perform the integration:
\[ W = K \int_{V_1}^{V_2} V^{-\gamma} \, dV \]
The integral of \(V^{-\gamma}\) is:
\[ \int V^{-\gamma} \, dV = \frac{V^{1-\gamma}}{1-\gamma} \]
Thus, we have:
\[ W = K \left[ \frac{V^{1-\gamma}}{1-\gamma} \right]_{V_1}^{V_2} \]
\[ W = \frac{K}{1-\gamma} \left( V_2^{1-\gamma} - V_1^{1-\gamma} \right) \]

### Step 5: Relate \(K\) to Pressure and Volume
Since \(K\) is constant, we can express it in terms of the initial and final states:
\[ K = P_1 V_1^\gamma = P_2 V_2^\gamma \]
Substituting this into the work equation gives:
\[ W = \frac{P_2 V_2^\gamma}{1-\gamma} \left( V_2^{1-\gamma} - V_1^{1-\gamma} \right) \]

### Step 6: Simplify the Expression
Using the relationship \(P_1 V_1^\gamma = P_2 V_2^\gamma\), we can further simplify:
\[ W = \frac{1}{1-\gamma} \left( P_2 V_2 - P_1 V_1 \right) \]

### Step 7: Relate Work Done to Temperature
Using the ideal gas law \(PV = nRT\), we can express the work done in terms of temperatures:
\[ W = \frac{nR}{1-\gamma} (T_2 - T_1) \]

### Final Result
Thus, the work done by the gas during the adiabatic process is:
\[ W = \frac{nR (T_2 - T_1)}{1 - \gamma} \]