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 can follow these steps:

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

### Step 2: Apply the Adiabatic Condition
Given that the volume is compressed to half its initial volume, we can denote the initial volume as \( V_1 \) and the final volume as \( V_2 = \frac{V_1}{2} \). According to the adiabatic condition:
\[ P_1 V_1^\gamma = P_2 V_2^\gamma \]

### Step 3: Substitute the Final Volume
Substituting \( V_2 \) into the equation:
\[ P_1 V_1^\gamma = P_2 \left(\frac{V_1}{2}\right)^\gamma \]
This simplifies to:
\[ P_1 V_1^\gamma = P_2 \frac{V_1^\gamma}{2^\gamma} \]

### Step 4: Rearranging the Equation
Rearranging the equation gives:
\[ P_2 = P_1 \cdot 2^\gamma \]
This shows how the pressure changes when the volume is halved.

### Step 5: Relate Pressure and Temperature
Using the ideal gas law, we can relate pressure, volume, and temperature:
\[ PV = nRT \]
From this, we can express temperature in terms of pressure and volume:
\[ T = \frac{PV}{nR} \]
Substituting \( P \) from the adiabatic relation into this equation will help us find a relationship involving \( T \).

### Step 6: Final Relationship
From the adiabatic condition and ideal gas law, we can derive:
\[ T V^{\gamma - 1} = \text{constant} \]
Substituting \(\gamma = \frac{5}{3}\):
\[ T V^{\frac{2}{3}} = \text{constant} \]

### Conclusion
Thus, the correct relationship for the adiabatic process of argon gas when compressed to half its volume is:
\[ T V^{\frac{2}{3}} = \text{constant} \]