Argon gas is adiabatically compressed to half its volume. If `P, V` and `T` represent the pressure, volume and temperature of the gasous, respectively, at any stage, then the correct equation representing the process is

Options

A. `TV^(2//5) =` constant
B. `TV^(5//3) =` constant
C. `TP^(-2//5) =` constant
D. `PT^(-2//5) =` constant

Explanation

To solve the problem of finding the correct equation representing the adiabatic compression of argon gas to half its volume, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Adiabatic Process**:
In an adiabatic process, there is no heat exchange with the surroundings. For a gas undergoing an adiabatic process, the relationship between pressure (P), volume (V), and temperature (T) can be described by the equation:
\[ PV^\gamma = \text{constant} \]
where \(\gamma\) (gamma) is the heat capacity ratio (C_p/C_v). For argon, which is a monoatomic gas, \(\gamma = \frac{5}{3}\).

2. **Apply the Ideal Gas Law**:
The ideal gas law states:
\[ PV = nRT \]
where \(n\) is the number of moles and \(R\) is the universal gas constant.

3. **Substitute Pressure in Terms of Volume and Temperature**:
From the ideal gas law, we can express pressure \(P\) as:
\[ P = \frac{nRT}{V} \]
Substitute this expression for \(P\) into the adiabatic equation:
\[ \left(\frac{nRT}{V}\right)V^\gamma = \text{constant} \]
Simplifying this gives:
\[ nRTV^{\gamma - 1} = \text{constant} \]

4. **Derive the Temperature-Volume Relationship**:
Rearranging the above equation leads to:
\[ TV^{\gamma - 1} = \text{constant} \]
This shows how temperature and volume are related during the adiabatic process.

5. **Determine the Specific Case of Volume Halving**:
Since the volume is compressed to half, we can denote the initial volume as \(V\) and the final volume as \(\frac{V}{2}\). Thus, substituting this into the equation gives:
\[ T\left(\frac{V}{2}\right)^{\gamma - 1} = \text{constant} \]

6. **Final Equation**:
The relationship can be expressed as:
\[ T \cdot V^{\gamma - 1} = \text{constant} \]
This indicates that as the volume decreases, the temperature will increase, consistent with the principles of adiabatic processes.

### Conclusion:
The correct equation representing the adiabatic compression of argon gas to half its volume is:
\[ TV^{\gamma - 1} = \text{constant} \]