If a gas of a volume `V_(1)` at pressure `p_(1)` is compressed adiabatically to volume `V_(2)` and pressure `p_(2)`, calculate the work done by the gas.

Options

A. zero
B. negative
C. positive
D. May be negative or positive

Answer

To calculate the work done by the gas during an adiabatic compression from volume \( V_1 \) at pressure \( p_1 \) to volume \( V_2 \) at pressure \( p_2 \), we can follow these steps:

### Step 1: Understand the Adiabatic Process
In an adiabatic process, there is no heat exchange with the surroundings. Therefore, the change in internal energy \( dQ \) is equal to the work done on the gas \( dW \).

### Step 2: Apply the First Law of Thermodynamics
The first law of thermodynamics states:
\[ dQ = dU + dW \]
For an adiabatic process, \( dQ = 0 \). Thus, we can simplify the equation to:
\[ 0 = dU + dW \implies dU = -dW \]
This indicates that the work done on the gas is equal to the decrease in internal energy.

### Step 3: Work Done in an Adiabatic Process
The work done \( W \) during an adiabatic process can be expressed in terms of the initial and final states of the gas. For an ideal gas undergoing an adiabatic process, the work done can be calculated using the formula:
\[ W = \frac{p_2 V_2 - p_1 V_1}{\gamma - 1} \]
where \( \gamma \) (gamma) is the heat capacity ratio \( C_p/C_v \).

### Step 4: Substitute the Values
Now, we can substitute the values of \( p_1, V_1, p_2, V_2 \), and \( \gamma \) into the equation to find the work done.

### Step 5: Conclusion
The work done by the gas during the adiabatic compression can be calculated using the derived formula. Note that since the gas is being compressed, the work done will be negative, indicating that work is done on the gas.

### Final Formula
Thus, the work done by the gas during the adiabatic process is:
\[ W = \frac{p_2 V_2 - p_1 V_1}{\gamma - 1} \]