Question

The temperature of a gas having \( 2.0 \times 10^{25} \) molecules per cubic meter at 1.38 atm (Given, \( k = 1.38 \times 10^{-23} \, \text{JK}^{-1} \)) is:

• A. 500 K
• B. 200 K
• C. 100 K
• D. 300 K

Answer 1

The Correct Option is A

To determine the temperature of the gas, we can use the Ideal Gas Law in terms of number of molecules. The formula is given by:

\(PV = NkT\) 

Where:

• \(P\) is the pressure of the gas.
• \(V\) is the volume.
• \(N\) is the number of molecules.
• \(k\) is the Boltzmann constant.
• \(T\) is the temperature in Kelvin.

The formula can be rearranged to solve for temperature \(T\):

\(T = \frac{PV}{Nk}\)

It is given:

• Pressure, \(P = 1.38 \, \text{atm}\). Converting to Pascal (since 1 atm = 101325 Pa): \(P = 1.38 \times 101325 \, \text{Pa}\)
• Number density, \(\frac{N}{V} = 2.0 \times 10^{25} \, \text{molecules per cubic meter}\)
• \(k = 1.38 \times 10^{-23} \, \text{JK}^{-1}\)

Substitute these values into the rearranged ideal gas formula:

\(T = \frac{1.38 \times 101325}{2.0 \times 10^{25} \times 1.38 \times 10^{-23}}\)

Calculate:

\(T = \frac{1.39657 \times 10^{5}}{2.76 \times 10^2}\)

\(T \approx 500 \, \text{K}\)

Thus, the temperature of the gas is 500 K.

The correct answer is: 500 K.

Answer 2

We use the ideal gas law in terms of the Boltzmann constant:  

\(PV = NkT\)

Where:  
- \( P = 1.38 \, \text{atm} = 1.38 \times 1.01 \times 10^5 \, \text{Pa} \),  
- \( N = 2.0 \times 10^{25} \) (total number of molecules),  
- \( k = 1.38 \times 10^{-23} \, \text{J K}^{-1} \).

Rearranging the formula to solve for \( T \):  
\(T = \frac{PV}{Nk}\)

Substituting the values:  

\(P = 1.38 \times 1.01 \times 10^5 = 1.01 \times 10^5 \, \text{Pa}\)

\(T = \frac{1.01 \times 10^5}{2 \times 10^{25} \times 1.38 \times 10^{-23}}\)
 

Simplifying, we get:  

\(T = \frac{1.01 \times 10^3}{2} \approx 500 \, \text{K}\)
Thus, the temperature \( T \) is 500 K.

The Correct Answer is: 500 K