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: Recall the ideal gas equation in molecular form.
For an ideal gas, the gas equation can be written in terms of number of molecules \( N \) as: \[ PV = Nk_BT \] where \( k_B \) is the Boltzmann constant.

Step 2: Distinguish it from the molar form.
The molar form of the ideal gas equation is: \[ PV = nRT \] But when the total number of molecules \( N \) is given instead of moles \( n \), we use: \[ PV = Nk_BT \]

Step 3: Identify the quantities in the question.
The problem directly gives: \[ N = \text{total number of molecules} \quad \text{and} \quad T = \text{temperature} \] Therefore the required expression should involve \( N \), \( k_B \), and \( T \).

Step 4: Write the required relation.
Thus, \[ PV = Nk_BT \] which may also be written as \[ PV = k_BNT \]

Step 5: Check dimensional consistency.
Boltzmann constant has dimensions: \[ [k_B] = \frac{\text{energy}}{\text{temperature}} \] So \[ k_B T = \text{energy} \] and multiplying by \( N \) gives total thermal energy scale, which is dimensionally consistent with \( PV \).

Step 6: Eliminate incorrect options.
Expressions like \( NT \), \( 2NT \), \( k_BT\sqrt{N} \), or \( k_BN\sqrt{T} \) do not match the ideal gas equation.
Only \[ k_BNT \] is correct.

Step 7: Final conclusion.
Hence, \[ \boxed{PV = k_BNT} \] Therefore, the correct option is \[ \boxed{(2)\ k_BNT} \]