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Newton calculated the speed of sound in air and got about 280 m/s. The measured value is about 331 m/s at 0 °C. Laplace closed the gap, and the reason is a thermodynamics idea hiding inside the waves chapter. This post explains both formulas, shows what the speed does and does not depend on, and works through examples with air, helium and a gas mixture.
Where is this in the syllabus?
NCERT's chapter Waves derives Newton's formula and the Laplace correction in Class 11. The JEE (Advanced) 2026 Information Brochure lists "Speed of sound in gases" and "Isothermal and adiabatic processes, bulk modulus of gases", and the Syllabus for JEE (Main) 2026 lists "speed of the travelling wave". Check the current year's syllabus before your exam.
What did Newton assume?
The speed of a longitudinal wave is v = √(B/ρ), where B is the bulk modulus of the medium and ρ its density. Newton assumed the compressions and rarefactions happen at constant temperature. For an isothermal change in an ideal gas, PV is constant, which gives B = P. So v = √(P/ρ).
NCERT's Example 14.4 puts in numbers for air at STP, a density of 1.29 kg/m³, and gets about 280 m/s. NCERT notes this is about 15% below the experimental value of 331 m/s.
What did Laplace correct?
As NCERT explains, Laplace pointed out that the pressure changes in a sound wave are so fast that heat has little time to flow and keep the temperature constant. The process is adiabatic, PV^γ = constant, which gives B = γP. So
v = √(γP/ρ)
For air, γ = 7/5. With the same numbers, v = 280 × √1.4 ≈ 332 m/s, close to NCERT's quoted 331.3 m/s; the small difference is rounding.
What does the speed depend on?
Use PV = μRT and ρ = μM/V, where M is the molar mass in kg/mol:
v = √(γRT/M)
This form answers NCERT's Exercise 14.4, which asks why the speed:
is independent of pressure: at fixed temperature P/ρ is constant, so changing the pressure alone does not change v
increases with temperature: v ∝ √T, with T in kelvin
increases with humidity: water vapour (about 18 g/mol) is lighter than air (about 29 g/mol), so moist air has a smaller average M
Worked examples
Take R = 8.314 J/(mol K), from the NIST CODATA value.
Air at 27 °C. v = √(1.4 × 8.314 × 300 / 0.029) ≈ √120 400 ≈ 347 m/s. Scaling from 0 °C gives the same: 331.3 × √(300/273) ≈ 347 m/s.
Helium at 0 °C. Helium is monatomic, so γ = 5/3, and M = 4.0 × 10⁻³ kg/mol. v = √(1.667 × 8.314 × 273 / 0.004) ≈ 972 m/s, nearly three times the speed in air.
When does the speed double? Since v ∝ √T, T must become four times larger: from 273 K to 1092 K, or 819 °C.
A mixture. Mix 1 mol of helium (Cv = 3R/2) with 1 mol of nitrogen (Cv = 5R/2):
Cv of the mixture = (1.5R + 2.5R)/2 = 2R, so Cp = 3R and γ = 1.5
M = (4 + 28)/2 = 16 g/mol
at 300 K, v = √(1.5 × 8.314 × 300 / 0.016) ≈ 484 m/s
Averaging the two γ values directly gives about 1.53, which is wrong; average Cv by moles instead.
Which mistakes cost marks?
Saying sound travels faster at higher pressure. At fixed temperature it does not.
Using °C inside √T. Convert to kelvin first.
Putting M in g/mol. In SI it must be in kg/mol, or the answer is off by a factor of √1000, about 31.6.
Using γ = 1.4 for every gas. It is 5/3 for monatomic gases such as helium and argon.
Applying Newton's formula when the question wants Laplace's, or the reverse. Read what is asked.
Why do sound questions catch students out?
Wittmann, Steinberg and Redish (2003) studied how university students reason about sound waves and found they often used sets of ideas different from the ones their teachers intended. Writing v = √(γRT/M) and asking what changes and what stays fixed keeps you on the model the question is testing.
A practice routine
Day 1: derive B = P and B = γP from PV = constant and PV^γ = constant, as NCERT does.
Day 2: five quick ratio problems on temperature, gas type and pressure, each answered in one line from v = √(γRT/M).
Day 3: one mixture problem and one humidity question.
Day 4: connect to the lab. Our post on the resonance tube experiment corrects a measured speed to 0 °C with the same √T rule.
If adiabatic and isothermal changes still feel abstract, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, each with its own lesson.
Today, explain to a friend in three sentences why Newton's answer came out about 15% low.
