Question

For an ideal gas at temperature \( T \) having the total number of molecules \( N \), the product of the pressure and volume, \( PV \), is equal to \((k_B \text{ is the Boltzmann constant})\)

• A. \( 2NT \)
• B. \( k_BNT \)
• C. \( k_BT\sqrt{N} \)
• D. \( k_BN\sqrt{T} \)
• E. \( NT \)

Hint:

Remember both forms of the ideal gas law: \[ PV=nRT \quad \text{and} \quad PV=Nk_BT \] Use \( nRT \) when moles are given, and \( Nk_BT \) when number of molecules is given.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question asks for the Ideal Gas Law equation expressed in terms of the number of molecules (\(N\)) and the Boltzmann constant (\(k_B\)), instead of the more common form using the number of moles (\(n\)) and the universal gas constant (\(R\)).
Step 2: Key Formula or Approach:
The standard form of the Ideal Gas Law is:
\[ PV = nRT \] We need to relate \(n\) and \(R\) to \(N\) and \(k_B\). The key relationships are:
1. Number of moles \(n = \frac{N}{N_A}\), where \(N\) is the total number of molecules and \(N_A\) is Avogadro's number.
2. The Boltzmann constant \(k_B\) is defined as the gas constant per molecule: \(k_B = \frac{R}{N_A}\). This can be rearranged to \(R = N_A k_B\).
Step 3: Detailed Explanation:
Start with the standard Ideal Gas Law:
\[ PV = nRT \] Substitute the expression for the number of moles, \(n = \frac{N}{N_A}\):
\[ PV = \left(\frac{N}{N_A}\right)RT \] Now, substitute the expression for the universal gas constant, \(R = N_A k_B\):
\[ PV = \left(\frac{N}{N_A}\right)(N_A k_B)T \] The Avogadro's number \(N_A\) in the numerator and denominator cancels out:
\[ PV = N k_B T \] This is the Ideal Gas Law in terms of the number of molecules. It is often written as k\(_B\)NT.
Step 4: Final Answer:
The product of pressure and volume, PV, for an ideal gas is equal to \(N k_B T\). This corresponds to option (B).