Ideal gas undergoes an adiabatic change in its state from `(P_1 V_1 , T_1)` to` (,P_2 ,V_2 T_2 )`. The work done (W) in the process is (n = number of moles, `C_p ` and `C_v` are molar specific heats of gas)

Options

A. `W=n (T_1 - T_2 ) C_P`
B. `W=n (T_1- T_2) C_v`
C. `W=n (T_1 +T_2 )C_p`
D. `W= n(T_1 +T_2 ) C_V`

Answer

To find the work done (W) during an adiabatic process for an ideal gas transitioning from state \((P_1, V_1, T_1)\) to \((P_2, V_2, T_2)\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Adiabatic Process**: In an adiabatic process, there is no heat exchange with the surroundings. This means that the change in internal energy (\( \Delta U \)) is equal to the work done on or by the system.

\[ \Delta U = W \]

2. **Use the First Law of Thermodynamics**: The first law of thermodynamics states that:

\[ \Delta U = Q - W \]

Since \(Q = 0\) for an adiabatic process, we have:

\[ \Delta U = -W \quad \Rightarrow \quad W = -\Delta U \]

3. **Calculate the Change in Internal Energy**: The change in internal energy for an ideal gas can be expressed as:

\[ \Delta U = n C_v (T_2 - T_1) \]

where \(n\) is the number of moles, and \(C_v\) is the molar specific heat at constant volume.

4. **Substituting into the Work Done Equation**: Substitute the expression for \(\Delta U\) into the work done equation:

\[ W = -n C_v (T_2 - T_1) \]

5. **Rearranging the Equation**: This can be rearranged to show that:

\[ W = n C_v (T_1 - T_2) \]

6. **Final Expression**: The work done by the gas during the adiabatic process can thus be expressed as:

\[ W = n C_v (T_1 - T_2) \]