Question 4 of 23intermediate🔧 ApplyNumerical3 marks

Estimate the speed of sound in air at standard temperature and pressure. The mass of 1 mole of air is 29.0×10−329.0 \times 10^{-3} kg.

Correct Answer

Step 1: Calculate density of air at STP

We know that 1 mole of any gas occupies 22.4 litres at STP. Therefore, density of air at STP is:

ρ = mass of one mole of air / volume of one mole of air at STP

ρ = (29.0×1029.0 \times 10⁻³ kg) / (22.4×1022.4 \times 10⁻³ m³) = 1.29 kg m⁻³

Step 2: Apply Newton's formula

According to Newton's formula, speed of sound in a gas is:

v = P/ρ\sqrt{P/ρ}

At STP, P = 1.01×101.01 \times 10⁵ N m⁻²

v = \sqrt{$$1.01 \times 10$$⁵ / 1.29} = 280 m s⁻¹

Step 3: Apply Laplace correction

Newton assumed isothermal conditions, but pressure variations during sound propagation are adiabatic. Using Laplace correction:

v = γP/ρ\sqrt{γP/ρ}

For air, γ = 7/5 = 1.4

v = γ×2802\sqrt{γ × 280²} = 1.4\sqrt{1.4} × 280 = 331.3 m s⁻¹

This agrees with the experimental value.

Exercise: EXAMPLES | Q: Example 14.4 | (Page 9)
For More Understanding

Explanation

The textbook demonstrates that Newton's isothermal assumption gives 280 m/s, which is 15% lower than the experimental value. Laplace corrected this by recognizing that sound propagation involves adiabatic (not isothermal) pressure variations, leading to the formula v = γP/ρ\sqrt{γP/ρ}. Using γ = 7/5 for air yields 331.3 m/s, matching experimental observations.

Solution Steps

  1. Step 1: Calculate density ρ = (29.0×1029.0 \times 10⁻³ kg)/(22.4×1022.4 \times 10⁻³ m³) = 1.29 kg m⁻³

  2. Step 2: Apply Newton's formula v = P/ρ\sqrt{P/ρ} = \sqrt{$$1.01 \times 10$$⁵/1.29} = 280 m s⁻¹

  3. Step 3: Apply Laplace correction v = γP/ρ\sqrt{γP/ρ} = \sqrt{$$1.4 \times 1.01$$ × 10⁵/1.29} = 331.3 m s⁻¹

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