Multiple choice

If $2:moles$ of diatomic gas and $1:mole$ of monatomic gas are mixed, then the ratio of specific heats for the mixture is

  1. $\displaystyle\frac{7}{3}$
  2. $\displaystyle\frac{5}{4}$
  3. $\displaystyle\frac{19}{13}$
  4. $\displaystyle\frac{15}{19}$
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C Correct answer
Explanation

For a mixture, the adiabatic index gamma_mix = (n1*Cv1 + n2*Cv2) / (n1*Cp1 + n2*Cp2) is not the standard formula. Use Cv_mix = (n1*Cv1 + n2*Cv2) / (n1+n2) and Cp_mix = (n1*Cp1 + n2*Cp2) / (n1+n2). For diatomic, Cv=5R/2, Cp=7R/2. For monatomic, Cv=3R/2, Cp=5R/2. Cv_mix = (2*2.5R + 1*1.5R)/3 = 6.5R/3. Cp_mix = (2*3.5R + 1*2.5R)/3 = 9.5R/3. Ratio = 9.5/6.5 = 19/13.

AI explanation

Using the mixture specific heat ratio formula gamma equals (n1 Cv1 plus n2 Cv2 plus total nR) divided by (n1 Cv1 plus n2 Cv2), we substitute the degrees of freedom values. For 1 mole of monatomic gas Cv is 3R/2 and for 2 moles of diatomic gas Cv is 5R/2, so the numerator becomes (1 times 3R/2) plus (2 times 5R/2) plus 3R, which equals 19R/2. The denominator is (1 times 3R/2) plus (2 times 5R/2), which equals 13R/2, making the ratio 19/13.

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