Question

A mixture of gasses consists of 16 g of Helium and 16 g of Oxygen. The ratio of specific heats of the mixture is nearly

• A. $1.33$
• B. $1.4$
• C. $1.56$
• D. $1.62$

Hint:

Mixture Specific Heat: \[ C_V,\textmix = \frac\sum n_i C_V,i\sum n_i,\quad \gamma = \fracC_PC_V \] Monoatomic: $C_V = \frac32R$, Diatomic: $C_V = \frac52R$

Answer 1

The Correct Option is D

Helium is monoatomic with molar mass $4 \text{ g/mol}$, so: \[ n_{He} = \frac{16}{4} = 4~\text{mol} \] Oxygen is diatomic with molar mass $32 \text{ g/mol}$, so: \[ n_{O_2} = \frac{16}{32} = 0.5~\text{mol} \] Using $C_V = \frac{3}{2}R$ for monoatomic and $\frac{5}{2}R$ for diatomic gases: \[ C_{V,\text{mix}} = \frac{4 \cdot \frac{3}{2}R + 0.5 \cdot \frac{5}{2}R}{4.5} = \frac{29R}{18} \] \[ C_{P,\text{mix}} = C_{V,\text{mix}} + R = \frac{29R}{18} + R = \frac{47R}{18} \] Thus: \[ \gamma_{\text{mix}} = \frac{C_P}{C_V} = \frac{47}{29} \approx 1.62 \] This matches option (4).

Answer 2

Step 1: Understand the problem and given data
- Mass of Helium (He) = 16 g
- Mass of Oxygen (O₂) = 16 g

Step 2: Find moles of each gas
- Molar mass of He = 4 g/mol
- Molar mass of O₂ = 32 g/mol
Moles of He = 16 / 4 = 4 mol
Moles of O₂ = 16 / 32 = 0.5 mol

Step 3: Specific heat ratios of individual gases
- For Helium (monatomic gas), γ₁ = 1.67
- For Oxygen (diatomic gas), γ₂ = 1.4

Step 4: Calculate mole fraction of each gas
Total moles = 4 + 0.5 = 4.5 mol
Mole fraction of He, x₁ = 4 / 4.5 ≈ 0.888
Mole fraction of O₂, x₂ = 0.5 / 4.5 ≈ 0.111

Step 5: Calculate specific heat at constant volume (C_v) and constant pressure (C_p) for the mixture
Using γ = C_p / C_v and the relation C_p - C_v = R (universal gas constant per mole), we can write:
C_v = R / (γ - 1)

Calculate C_v for each gas:
C_v(He) = R / (1.67 - 1) = R / 0.67 ≈ 1.4925 R
C_v(O₂) = R / (1.4 - 1) = R / 0.4 = 2.5 R

Calculate mixture C_v:
C_v(mix) = x₁ × C_v(He) + x₂ × C_v(O₂)
= 0.888 × 1.4925 R + 0.111 × 2.5 R ≈ 1.325 R + 0.278 R = 1.603 R

Calculate mixture C_p:
C_p(mix) = C_v(mix) + R = 1.603 R + R = 2.603 R

Step 6: Calculate γ for the mixture
γ(mix) = C_p(mix) / C_v(mix) = 2.603 R / 1.603 R ≈ 1.62

Step 7: Conclusion
The ratio of specific heats for the gas mixture is approximately 1.62.