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 need to analyze the relationship between the volume (V) and temperature (T) of an ideal gas under constant pressure using the ideal gas law. Let's break it down step by step:

### Step 1: Understand the Ideal Gas Law
The ideal gas law is given by the equation:
\[ PV = nRT \]
Where:
- \( P \) = Pressure
- \( V \) = Volume
- \( n \) = Number of moles
- \( R \) = Ideal gas constant
- \( T \) = Temperature in Kelvin

### Step 2: Rearranging the Ideal Gas Law
Since the pressure (P) is constant, we can rearrange the ideal gas law to express volume (V) in terms of temperature (T):
\[ V = \frac{nRT}{P} \]

### Step 3: Taking the Logarithm
To find the relationship between log V and log T, we take the logarithm of both sides:
\[ \log V = \log\left(\frac{nRT}{P}\right) \]
Using the properties of logarithms, this can be expanded to:
\[ \log V = \log(nR) - \log(P) + \log(T) \]

### Step 4: Identifying the Linear Relationship
This can be rewritten as:
\[ \log V = \left(\log(nR) - \log(P)\right) + \log(T) \]
This shows that the graph of \( \log V \) vs. \( \log T \) is a straight line with:
- Slope (m) = 1
- Y-intercept (c) = \( \log(nR/P) \)

### Step 5: Calculating the Y-intercept
Given:
- \( n = 10 \) moles
- \( R = 0.0821 \, \text{L atm/(K mol)} \)
- \( P = 0.821 \, \text{atm} \)

We can calculate the intercept:
\[ \log\left(\frac{nR}{P}\right) = \log\left(\frac{10 \times 0.0821}{0.821}\right) \]

Calculating the value inside the logarithm:
\[ \frac{10 \times 0.0821}{0.821} = 1 \]
Thus:
\[ \log(1) = 0 \]

### Step 6: Conclusion of the Graph
The equation of the line is:
\[ \log V = 0 + 1 \cdot \log T \]
This indicates that:
- The slope is 1 (which corresponds to a 45-degree angle).
- The intercept is 0.

### Step 7: Analyzing the Graph Options
From the analysis, we can conclude:
- The correct graph must have a slope of 1 (45 degrees) and an intercept of 0.
- Therefore, options with intercepts not equal to 0 or slopes not equal to 1 are incorrect.

### Final Answer
The correct option representing the variation of \( \log V \) vs. \( \log T \) is **Option 1**.

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