A uniform tube of volume V contains an ideal monatomic gas at a uniform pressure `P` the temperature of the gas varies linearly along te length of the tube ad T and 2 T are the temperature at the ends.

Options

A. The number of moles of the gas in the tube is `(PV)/(RT)l n2`
B. The number of moles of the gas in the tube is `(PV)/(RT) l n4`
C. The internal energy of the gas in the tube is `(3//2)PV`
D. The internal energy of the gas in the tube is `(3//2)nRT`.

Explanation

To solve the problem, we need to determine the number of moles of the ideal monatomic gas in the tube and the internal energy of the gas. Let's break this down step by step.

### Step 1: Understand the Temperature Variation
The problem states that the temperature varies linearly along the length of the tube, with temperatures \( T \) at one end and \( 2T \) at the other end.

### Step 2: Relate Temperature to Length
Let \( L \) be the length of the tube. The temperature \( T' \) at a distance \( x \) from the end with temperature \( T \) can be expressed as:
\[ T' = T + \left(\frac{2T - T}{L}\right)x = T + \frac{T}{L}x \]
This simplifies to:
\[ T' = T + \frac{T}{L}x \]

### Step 3: Ideal Gas Law
Using the ideal gas equation:
\[ PV = nRT \]
We can express the number of moles \( n \) as:
\[ n = \frac{PV}{RT} \]
For a small section of the tube of length \( dx \), the number of moles \( dN \) in that section can be expressed as:
\[ dN = \frac{P \cdot A \cdot dx}{R \cdot T'} \]
where \( A \) is the cross-sectional area of the tube.

### Step 4: Substitute for \( dx \)
From the temperature equation, we can differentiate to find \( dx \) in terms of \( dT' \):
\[ \frac{dT'}{dx} = \frac{T}{L} \implies dx = \frac{L}{T} dT' \]

### Step 5: Substitute \( dx \) into the Number of Moles Equation
Substituting \( dx \) into the expression for \( dN \):
\[ dN = \frac{P \cdot A}{R \cdot T'} \cdot \frac{L}{T} dT' \]
This simplifies to:
\[ dN = \frac{P \cdot A \cdot L}{R \cdot T} \cdot \frac{1}{T'} dT' \]

### Step 6: Integrate to Find Total Moles
To find the total number of moles \( N \), we integrate \( dN \) from \( T \) to \( 2T \):
\[ N = \int_{T}^{2T} \frac{P \cdot A \cdot L}{R \cdot T} \cdot \frac{1}{T'} dT' \]
This integral evaluates to:
\[ N = \frac{P \cdot A \cdot L}{R \cdot T} \ln\left(\frac{2T}{T}\right) = \frac{P \cdot A \cdot L}{R \cdot T} \ln(2) \]

### Step 7: Relate Volume to Area and Length
Since \( V = A \cdot L \), we can substitute \( A \cdot L \) with \( V \):
\[ N = \frac{PV}{RT} \ln(2) \]

### Step 8: Calculate Internal Energy
The internal energy \( U \) of an ideal monatomic gas is given by:
\[ U = \frac{3}{2} nRT \]
Substituting \( n \) from our previous result:
\[ U = \frac{3}{2} \left(\frac{PV}{RT} \ln(2)\right) RT = \frac{3}{2} PV \ln(2) \]

### Final Results
1. The number of moles of the gas is:
\[ N = \frac{PV}{RT} \ln(2) \]
2. The internal energy of the gas is:
\[ U = \frac{3}{2} PV \ln(2) \]