Speed of Sound (Ideal Gas) Calculator
Solve the ideal-gas speed of sound a = √(γ·R·T) for any of its three unknowns — speed of sound, temperature, or ratio of specific heats — with step-by-step math, a live wavefront schematic, and full companion acoustic metrics.
T ↑ — a grows as √T
Quadrupling the absolute temperature only doubles the speed of sound. Air at 273 K carries sound at 331 m/s; at 1093 K (four times as hot) it is only 663 m/s. The square root makes temperature a surprisingly weak lever.
Pressure has no effect
Because p = ρRT, compressing a gas raises its density by exactly the same factor that it raises pressure. The two effects cancel exactly in a = √(γ·R·T), so the speed of sound is independent of pressure — a purely ideal-gas result.
Light gases carry sound faster
Helium has R = 2077 J/(kg·K), roughly seven times that of air, and γ = 1.667 versus 1.4. The combination makes sound travel almost three times faster in helium — 1007 m/s at 20 °C versus 343 m/s in air. That is why voices turn squeaky after breathing helium.
γ enters through stiffness
The ratio γ = c_p/c_v measures how stiff the gas is under rapid (isentropic) compression. A monatomic gas has γ = 1.667 and is stiffer per unit density than a polyatomic one with γ ≈ 1.3, so it transmits disturbances more quickly for the same R and T.
How to Use This Calculator
The ideal-gas speed-of-sound relation links three quantities: the speed of sound a, the absolute temperature T, and the ratio of specific heats γ, with R as a gas property. Pick which one you want the tool to solve, then fill in the other two.
1. Choose what to solve for
Pick a (speed of sound), T (temperature) or γ (specific-heat ratio) — the tool rearranges a = √(γ·R·T) automatically for whichever you choose.
2. Pick the gas
Use a preset (air, N₂, O₂, He, Ar, CO₂, CH₄, H₂O, H₂) to set both γ and R at once, or enter your own values for a custom gas mixture.
3. Read the result and steps
The result card shows the solved value plus engineering notation and the same speed in km/h and knots. The step-by-step panel shows exactly how the number was reached.
4. Check the schematic and chart
The live wavefront schematic widens or tightens the spacing as the speed of sound changes, and the chart plots a against T for your gas alongside air and helium.
5. Switch to Advanced for depth
Advanced mode adds the a vs. γ curve at fixed T and R, a sensitivity check on a 1% input change, and companion metrics including mean molecular speed and wavelength at 1 kHz.
6. Convert units and cross-check
Use the temperature and speed converters further down the page to switch between K/°C/°F/°R and m/s/km/h/mph/kn/ft/s, or "Copy result" to grab the solved value.
What This Tool Offers
Built around the ideal-gas relation a = √(γ·R·T), the starting point for acoustics, gas dynamics and compressible-flow analysis.
Solve any of three unknowns
Go from temperature to speed of sound, recover temperature from a measured speed, or identify the effective γ of an unknown gas.
Nine gas presets
Air, N₂, O₂, He, Ar, CO₂, CH₄, H₂O and H₂ — each sets both γ and R in a single click, with a full custom override.
Live wavefront schematic
A source emitting expanding wavefronts whose spacing tracks the speed of sound, with the wavelength at 1 kHz called out.
Comparison charts
a vs. T for your gas alongside air and helium, plus an advanced a vs. γ curve showing the mild square-root dependence.
Companion acoustic metrics
Mean molecular speed √(8RT/π), its ratio to a, wavelength at 1 kHz, and the speed of sound in km/h, mph, knots and ft/s.
Reference tables
Speed of sound in nine common gases at 20 °C, in air across a wide temperature span, and a γ / R reference table.
Reference Tables & Conversions
Temperature and speed unit converters, plus lookup tables for the speed of sound in common gases and in air at various temperatures.
Temperature converter
Speed converter
Speed of Sound in Common Gases at 20 °C (293.15 K)
Every value below is computed from a = √(γ·R·T) with the γ and R listed in the reference table further down.
| Gas | γ | R (J/(kg·K)) | a at 20 °C (m/s) | a at 20 °C (km/h) |
|---|
Speed of Sound in Air vs. Temperature
Computed for dry air with γ = 1.40 and R = 287.05 J/(kg·K). At high temperatures real air dissociates and γ falls, so the top rows are ideal-gas extrapolations.
| T (°C) | T (K) | a (m/s) | a (km/h) | a (mph) |
|---|
Gas Property Reference
| Gas | Molar mass M (g/mol) | γ = c_p/c_v | R = 8314.46 / M | Typical use |
|---|---|---|---|---|
| Hydrogen (H₂) | 2.016 | 1.410 | 4124.2 | Lightest gas — fastest sound propagation |
| Helium (He) | 4.003 | 1.667 | 2077.1 | Monatomic; high γ, high R |
| Steam (H₂O) | 18.015 | 1.330 | 461.5 | Polyatomic; γ falls with temperature |
| Nitrogen (N₂) | 28.013 | 1.400 | 296.8 | Diatomic; dominant component of air |
| Air (dry) | 28.964 | 1.400 | 287.05 | Standard reference gas |
| Oxygen (O₂) | 31.999 | 1.395 | 259.8 | Diatomic |
| Argon (Ar) | 39.948 | 1.667 | 208.1 | Monatomic; heavy noble gas |
| Carbon dioxide (CO₂) | 44.010 | 1.289 | 188.9 | Polyatomic; low γ, low R |
| Methane (CH₄) | 16.043 | 1.310 | 518.3 | Polyatomic fuel gas |
What Does a = √(γ·R·T) Measure?
The speed of sound is the speed at which a small pressure disturbance travels through a compressible medium. In an ideal gas it is set entirely by how stiff the gas is (measured by γ) and how warm it is (measured by T), scaled by the gas's specific gas constant R. It does not depend on pressure, on density, or on the volume of gas you happen to be considering — a result that surprises most people the first time they meet it.
Speed of Sound Formula
The defining relationship for an ideal gas is:
Rearranged to solve for the other two unknowns:
γ = a² / ( R · T )
The quantity R is the specific gas constant, R = R_universal / M, where R_universal = 8314.46 J/(kmol·K) and M is the molar mass in kg/kmol. For dry air, M = 28.964 kg/kmol, so R = 8314.46 / 28.964 = 287.05 J/(kg·K).
Speed of Sound Variables
| Symbol | Meaning | Notes |
|---|---|---|
| a | Speed of sound | Always positive; independent of pressure |
| γ | Ratio of specific heats, c_p/c_v | 1.40 for air, 1.667 for monatomic gases |
| R | Specific gas constant, R_univ / M | 287.05 J/(kg·K) for dry air |
| T | Absolute (thermodynamic) temperature | Must be in kelvin, never °C or °F |
| M | Molar mass | 28.964 g/mol for dry air |
| R_univ | Universal gas constant | 8314.46 J/(kmol·K) |
How Temperature and Gas Affect the Speed of Sound
Because a is proportional to √T, temperature is a weaker lever than most people expect. Warming air from 0 °C to 40 °C — a 15% rise in absolute temperature — raises the speed of sound by only about 7%, from 331 m/s to 355 m/s. Doubling the absolute temperature raises a by only 41%.
Changing the gas is far more dramatic. Helium has roughly seven times the specific gas constant of air and a higher γ, so sound travels at about 1007 m/s in helium at 20 °C — nearly three times faster than the 343 m/s in air. This is why helium balloons make voices sound high-pitched: the resonance frequencies of the vocal tract scale with the speed of sound.
Speed of Sound Calculation Examples
γ · R · T = 1.40 × 287.05 × 293.15 = 117,808.2
a = √117,808.2 = 343.23 m/s
a² = 343.23² = 117,808.2
γ · R = 1.40 × 287.05 = 401.87
T = 117,808.2 / 401.87 = 293.15 K (= 20.0 °C)
a² = 1007² = 1,014,049
R · T = 2077.1 × 293.15 = 608,902
γ = 1,014,049 / 608,902 = 1.665 ≈ 1.667 (helium)
Frequently Asked Questions
Because the kinetic energy of the gas molecules is proportional to the absolute temperature, while the speed of the molecules is proportional to the square root of kinetic energy. Sound is transmitted by molecular collisions, so its propagation speed scales with molecular speed — hence √T. To double the speed of sound you would need to quadruple the absolute temperature.
For an ideal gas, p = ρRT. If you double the pressure at constant temperature, you also double the density. The stiffness of the gas rises, but so does its inertia, and the two effects cancel exactly. The result is that a = √(γ·R·T) contains no pressure term at all. In a real gas at very high pressure, intermolecular forces break this cancellation and the speed does begin to depend slightly on pressure.
Two reasons. First, helium is monatomic, so its γ is 1.667 versus 1.40 for the diatomic air molecules — a stiffer response to compression. Second, and more importantly, helium's molar mass is only 4 g/mol versus 28.96 for air, so its specific gas constant R is about seven times larger. The combination gives a = 1007 m/s in helium at 20 °C compared with 343 m/s in air.
Compute the mixture's effective molar mass M_mix by mole-fraction weighting, then use R = 8314.46 / M_mix. For γ, use a mole-fraction-weighted average of the individual γ values as a first approximation, or look up the mixture γ from a thermodynamic table for better accuracy. For dry air, M_mix = 28.964 g/mol gives R = 287.05 J/(kg·K), and γ = 1.40 is standard up to about 500 K.
Yes, slightly. Water vapour has a molar mass of 18.015 g/mol versus 28.964 for dry air, so humid air is less dense and has a higher effective R. At 20 °C, going from perfectly dry air to saturated air raises the speed of sound by roughly 0.4%, from about 343.2 m/s to 344.5 m/s. For most engineering work at ordinary conditions this is neglected.
The mean molecular speed is v̄ = √(8RT/π), which is always larger than a. The ratio is v̄ / a = √(8/(πγ)), which is about 1.35 for air. So at 20 °C, air molecules move at roughly 463 m/s on average while sound propagates at 343 m/s. Sound is a collective pressure wave, not a single molecule travelling from source to listener.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| a | Speed of sound | m/s |
| γ | Ratio of specific heats | |
| R | Specific gas constant | J/(kg*K) |
| T | Static temperature | K |
What it means
The speed of sound is the speed at which small pressure disturbances propagate through a medium. In an ideal gas, it depends only on the temperature and gas properties, not on pressure. This formula is derived from the isentropic compressibility. It is fundamental in aerodynamics, as it defines the Mach number and influences compressibility effects. The speed of sound decreases with altitude (lower temperature), which affects aircraft performance. It is also used in meteorology and acoustics. Understanding this relation is essential for any compressible flow analysis and for interpreting flight data.
Worked example
Speed of Sound – Two Examples
Real‑World| Parameter | Value |
|---|---|
| γ | 1.4 |
| R | 287 J/kg·K |
| T | 288 K |
| Parameter | Value |
|---|---|
| T | 216.5 K |
Common mistakes
- Speed of sound (ideal gas): a = √(γ·R·T).
- γ: Specific heat ratio (dimensionless).
- R: Specific gas constant (J/(kg·K)).
- T: Absolute temperature (K).
- Units: (m²/s²) → m/s.
Applications
The speed of sound in an ideal gas, a = √(γ·R·T), is a fundamental property that determines the Mach number. It varies with temperature; higher temperatures increase the speed of sound. Engineers use this formula to compute Mach numbers, to design acoustic liners, and to assess the effects of altitude on engine performance. In propulsion, the speed of sound affects the design of turbomachinery (compressor blade speeds). By understanding the speed of sound, aerospace engineers can interpret compressible flow phenomena and ensure that aircraft operate within the desired Mach regime.
- Mach number calculation for flight test and design
- Engine component design (compressor, turbine tip speeds)
- Noise generation and acoustic propagation studies
- High‑speed aerodynamic testing and instrumentation
- Atmospheric property modelling for flight planning
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