For a gas undergoing an adiabatic change, the relation between temperature and volume is found to be `TV^(0.4)` = constant. This gas must be

Options

A. Helium
B. Carbon dioxide
C. Argon
D. Hydrogen

Explanation

To solve the problem, we need to analyze the given relationship between temperature (T) and volume (V) for a gas undergoing an adiabatic process. The relationship is given by:

\[ TV^{0.4} = \text{constant} \]

### Step-by-Step Solution:

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) can be described using the adiabatic condition, which states:
\[ PV^{\gamma} = \text{constant} \]
where \(\gamma\) is the heat capacity ratio (specific heat ratio).

2. **Relate the Given Equation to the Adiabatic Condition**:
We are given the equation:
\[ TV^{0.4} = \text{constant} \]
We can rewrite this in terms of the adiabatic condition. From the adiabatic condition, we can express it in terms of temperature and volume:
\[ TV^{\gamma - 1} = \text{constant} \]

3. **Compare the Exponents**:
From the two equations, we can compare the exponents of V:
\[ \gamma - 1 = 0.4 \]
Solving for \(\gamma\):
\[ \gamma = 0.4 + 1 = 1.4 \]

4. **Identify the Type of Gas**:
The value of \(\gamma\) depends on the type of gas. For different types of gases:
- Monatomic gases (e.g., Helium): \(\gamma \approx 1.67\)
- Diatomic gases (e.g., Oxygen, Nitrogen, Hydrogen): \(\gamma \approx 1.4\)
- Polyatomic gases (e.g., Carbon Dioxide): \(\gamma < 1.4\)

Since we found \(\gamma = 1.4\), this indicates that the gas is a diatomic gas.

5. **Choose the Correct Gas**:
Among the options provided:
- Helium (Monatomic)
- Carbon Dioxide (Polyatomic)
- Argon (Monatomic)
- Hydrogen (Diatomic)

The only diatomic gas listed is Hydrogen.

### Final Answer:
The gas must be **Hydrogen gas**.