Two identical vessels A and B contain masses m and 2m of same gas. The gases in the vessels are heated keeping their volumes constant and equal. The temperature-pressure curve for mass 2m makes angle `alpha` with T-axis and that and for mass m makes an angle `beta` with T-axis then

Options

A. `tanalpha=tan beta`
B. `tan alpha =2 tan beta`
C. `tan beta=2 tan alpha`
D. `None of the Above

Answer

To solve the problem, we need to analyze the relationship between the temperature and pressure of the gases in the two vessels using the ideal gas law and the characteristics of the temperature-pressure curve.

### Step-by-Step Solution:

1. **Understanding the Ideal Gas Law**:
The ideal gas law is given by the equation:
\[ PV = nRT \]
where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature.

2. **Rearranging the Ideal Gas Law**:
Since the volumes of the vessels are constant and equal, we can express the relationship between temperature and pressure as:
\[ \frac{T}{P} = \frac{V}{nR} \]
This shows that the ratio \( \frac{T}{P} \) is dependent on the number of moles \( n \).

3. **Relating Moles to Mass**:
The number of moles \( n \) can be expressed in terms of mass \( m \) and molar mass \( M \):
\[ n = \frac{m}{M} \]
Therefore, we can rewrite the equation as:
\[ \frac{T}{P} = \frac{V \cdot M}{mR} \]

4. **Identifying the Slopes**:
The slope of the temperature-pressure curve (which is \( \frac{T}{P} \)) is inversely related to the mass of the gas. This means that:
- For vessel A (mass \( m \)), the slope is \( \tan(\beta) \).
- For vessel B (mass \( 2m \)), the slope is \( \tan(\alpha) \).

5. **Setting Up the Relationship**:
Since the slopes are inversely related to the masses, we can write:
\[ \tan(\alpha) \propto \frac{1}{2m} \quad \text{and} \quad \tan(\beta) \propto \frac{1}{m} \]
Thus, we can express the relationship between the slopes:
\[ \frac{\tan(\beta)}{\tan(\alpha)} = \frac{2m}{m} = 2 \]

6. **Final Expression**:
From the above relationship, we can conclude:
\[ \tan(\beta) = 2 \tan(\alpha) \]

### Conclusion:
The relationship between the angles is given by:
\[ \tan(\beta) = 2 \tan(\alpha) \]