The temperature of a gas having 2.0 10^25 molecules per cubic meter at 1.38 atm (Given, k = 1.38 10^-23 J K^-1) is:

Given: number density NV = 2.0 10^25 m^-3, pressure P = 1.38 atm, and Boltzmann constant k = 1.38 10^-23 J K^-1. Find: the temperature T of the gas. Use the ideal gas law in molecular form: PV = NkT Since the number of molecules per unit volume is given, write P = NVkT So, T = P ( NV)k Convert pressure into pascal: P = 1.38 101325 Pa Substitute the values: T = 1.38 1013252.0 10^25 1.38 10^-23 Now simplify: T = 1.39657 10^52.76 10^2 T 500 K Therefore, the temperature of the gas is 500 K. The correct option is A.

The temperature of a gas having 2.0 10^25 molecules per cubic met… — Solution | JEE Physics

MCQEasyJEE 2024Equation of State of Ideal Gas

The temperature of a gas having $2.0\times10^{25}$ molecules per cubic meter at $1.38\,\text{atm}$ (Given, $k = 1.38\times10^{-23}\,\text{J K}^{-1}$ ) is:

A
$500\,\text{K}$
B
$200\,\text{K}$
C
$100\,\text{K}$
D
$300\,\text{K}$

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: number density $\frac{N}{V} = 2.0\times10^{25}\,\text{m}^{-3}$ , pressure $P = 1.38\,\text{atm}$ , and Boltzmann constant $k = 1.38\times10^{-23}\,\text{J K}^{-1}$ .

Find: the temperature $T$ of the gas.

Use the ideal gas law in molecular form:

PV = NkT

Since the number of molecules per unit volume is given, write

P = \frac{N}{V}kT

So,

T = \frac{P}{\left(\frac{N}{V}\right)k}

Convert pressure into pascal:

P = 1.38 \times 101325\,\text{Pa}

Substitute the values:

T = \frac{1.38 \times 101325}{2.0\times10^{25} \times 1.38\times10^{-23}}

Now simplify:

T = \frac{1.39657\times10^{5}}{2.76\times10^{2}}

T \approx 500\,\text{K}

Therefore, the temperature of the gas is $500\,\text{K}$ . The correct option is A.

Common mistakes

Using $PV = nRT$ directly with number of molecules is incorrect because $n$ is in moles, not molecules. Use $PV = NkT$ or convert molecules into moles first.
Treating $2.0\times10^{25}$ as the total number of molecules instead of number density is wrong. The given quantity is molecules per cubic meter, so use $\frac{N}{V}$ in the formula.
Not converting pressure from atmosphere to pascal leads to a wrong numerical value. Replace $1.38\,\text{atm}$ by $1.38\times101325\,\text{Pa}$ before substitution.

Practice more Equation of State of Ideal Gas questions

Get unlimited AI-adaptive practice, mastery tracking, and an AI tutor that explains every step — free to start.

Start free trial More on Equation of State of Ideal Gas

Answer

Step-by-step solution

Common mistakes

Practice more Equation of State of Ideal Gas questions

Related questions