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Heat and Thermodynamics question

2022 · 25 Jul · Shift 2 · Q57
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  5. /2022 · 25 Jul · Shift 2 · Q57

Heat and Thermodynamics question

2022 · 25 Jul · Shift 2 · Q57

JEE MainPhysicsHeat and ThermodynamicsMCQ+4 / −1
Sound travels in a mixture of two moles of helium and n moles of hydrogen. If rms speed of gas molecules in the mixture is 2\sqrt22​ times the speed of sound, then the value of n will be :
  1. A
    1
  2. B
    2
  3. C
    3
  4. D
    4
View written solutionFree

Correct answer: B

  1. Given relation between RMS speed and speed of sound

For an ideal gas mixture:

  • RMS speed of molecules: vrms=3RTMv_{\text{rms}}=\sqrt{\frac{3RT}{M}}vrms​=M3RT​​
  • Speed of sound: vs=γRTMv_s=\sqrt{\frac{\gamma RT}{M}}vs​=MγRT​​

Here, MMM is the mean molar mass of the gas mixture and γ\gammaγ is the effective ratio of specific heats of the mixture.

Given: vrms=2 vsv_{\text{rms}}=\sqrt{2}\,v_svrms​=2​vs​

Squaring both sides: 3RTM=2⋅γRTM\frac{3RT}{M}=2\cdot \frac{\gamma RT}{M}M3RT​=2⋅MγRT​

So, 3=2γ3=2\gamma3=2γ γ=32\gamma=\frac{3}{2}γ=23​

  1. Find effective γ\gammaγ of the mixture

The mixture contains:

  • Helium: 222 moles
  • Hydrogen: nnn moles

For helium (monatomic): CV,He=3R2,CP,He=5R2C_{V,He}=\frac{3R}{2},\qquad C_{P,He}=\frac{5R}{2}CV,He​=23R​,CP,He​=25R​

For hydrogen (diatomic): CV,H2=5R2,CP,H2=7R2C_{V,H_2}=\frac{5R}{2},\qquad C_{P,H_2}=\frac{7R}{2}CV,H2​​=25R​,CP,H2​​=27R​

Total heat capacities of the mixture:

=3R+\frac{5nR}{2}$$ $$C_P=2\left(\frac{5R}{2}\right)+n\left(\frac{7R}{2}\right) =5R+\frac{7nR}{2}$$ Thus, $$\gamma=\frac{C_P}{C_V}=\frac{5R+\frac{7nR}{2}}{3R+\frac{5nR}{2}}$$ Cancel $R$: $$\gamma=\frac{5+\frac{7n}{2}}{3+\frac{5n}{2}}$$ Multiply numerator and denominator by $2$: $$\gamma=\frac{10+7n}{6+5n}$$ --- 3. **Use the condition $\gamma=\frac{3}{2}$** $$\frac{10+7n}{6+5n}=\frac{3}{2}$$ Cross-multiplying: $$2(10+7n)=3(6+5n)$$ $$20+14n=18+15n$$ $$n=2$$ --- 4. **Check with options** The correct option is: $$\boxed{2}$$ So, **Option B** is correct.
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