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

The temperature of the gas can be calculated using the Ideal Gas Law expressed in terms of the number of molecules. The relevant equation is:

\(PV = NkT\)

Where:

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

To find the temperature \(T\), the formula is rearranged as follows:

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

The following values are provided:

• Pressure: \(P = 1.38 \, \text{atm}\). This is converted to Pascals (1 atm = 101325 Pa), resulting in: \(P = 1.38 \times 101325 \, \text{Pa}\).
• Number density: \(\frac{N}{V} = 2.0 \times 10^{25} \, \text{molecules per cubic meter}\).
• Boltzmann constant: \(k = 1.38 \times 10^{-23} \, \text{JK}^{-1}\).

These values are substituted into the rearranged ideal gas law equation:

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

Performing the calculation yields:

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

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

Therefore, the gas temperature is 500 K.

The correct answer is: 500 K.