Ten moles of an ideal gas are filled in a closed vessel. The vessel has cylinder and piston type arrangement and pressure of the gas remains constant at 0.821 atm. Which of the following graph represents correct variation of log V vs log T?
(V = Volume in litre and T = temperature in Kelvin)

Options

A.
B.
C.
D.

Answer

To solve the problem, we will use the ideal gas law and analyze the relationship between volume (V) and temperature (T) for an ideal gas at constant pressure.

### Step-by-Step Solution:

1. **Understand the Ideal Gas Law**:
The ideal gas law is given by the equation:
\[ PV = nRT \]
where:
- \( P \) = pressure (in atm)
- \( V \) = volume (in liters)
- \( n \) = number of moles of gas
- \( R \) = ideal gas constant (0.0821 L·atm/(K·mol))
- \( T \) = temperature (in Kelvin)

2. **Substitute Known Values**:
We know:
- \( n = 10 \) moles
- \( P = 0.821 \) atm
- \( R = 0.0821 \) L·atm/(K·mol)

Plugging these values into the ideal gas equation, we have:
\[ 0.821V = 10 \times 0.0821 \times T \]

3. **Rearranging the Equation**:
Rearranging the equation to solve for \( V \):
\[ V = \frac{10 \times 0.0821 \times T}{0.821} \]
Simplifying this gives:
\[ V = \frac{0.821 \times 10 \times 0.0821}{0.821} \times T = 10 \times 0.0821 \times T \]
Thus, we can express \( V \) as:
\[ V = kT \]
where \( k = \frac{10 \times 0.0821}{0.821} \).

4. **Taking the Logarithm**:
Now, we take the logarithm of both sides:
\[ \log V = \log(kT) \]
Using the property of logarithms:
\[ \log V = \log k + \log T \]

5. **Rearranging the Logarithmic Equation**:
This can be rearranged to:
\[ \log V = \log T + \log k \]
This equation is in the form of \( y = mx + b \), where:
- \( y = \log V \)
- \( x = \log T \)
- \( m = 1 \) (slope)
- \( b = \log k \) (y-intercept)

6. **Graph Characteristics**:
Since the slope \( m = 1 \), the graph of \( \log V \) versus \( \log T \) will be a straight line with a slope of 1, passing through the y-intercept \( \log k \). This indicates that the graph will pass through the origin if \( k = 1 \).

7. **Identifying the Correct Graph**:
The graph of \( \log V \) vs. \( \log T \) will be a straight line with a slope of 1, indicating a 45-degree angle with respect to the axes. Therefore, the correct graph will be the one that represents a straight line passing through the origin with a slope of 1.

### Conclusion:
The correct option is the one that shows a straight line passing through the origin with a slope of 1 (45 degrees). This corresponds to option 1.