At what temperatures (in $^{\circ}$C) will the speed of sound in air be 3 times its value at $0^{\circ}$C?

At what temperatures (in °C) will the speed of sound in air be 3 times its value at 0°C?

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Concept understanding — Speed of Sound in Gases

Speed of Sound in Gases – From Intuition to Precision

Imagine you're standing at one end of a long, empty hallway. Your friend is at the other end. When you clap your hands, the sound doesn't reach them instantly — it takes a small but noticeable fraction of a second. That delay is the speed of sound in air.

Now think about why sound travels at all. Sound is a mechanical wave — it needs a medium (like air, water, or steel) to travel. When you clap, you push the air molecules near your hands. Those molecules bump into their neighbours, which bump into the next ones, and so on. This chain of collisions carries the disturbance forward. The speed at which this "bump" travels depends on two things:

How stiff the medium is — how quickly it resists being compressed.
How heavy the medium is — how much inertia each molecule has.

In a gas, both of these are linked to temperature and the gas's molecular properties.

The Precise Statement

For an ideal gas, the speed of sound $v$ is given by:

v = \sqrt{\frac{\gamma R T}{M}}

Where:

$\gamma$ (gamma) is the adiabatic index — the ratio of specific heats $C_p / C_v$ . For air (mostly diatomic gases like N₂ and O₂), $\gamma \approx 1.4$ .
$R$ is the universal gas constant ( $8.314\ \text{J/mol·K}$ ).
$T$ is the absolute temperature in Kelvin.
$M$ is the molar mass of the gas (in kg/mol).

v = \sqrt{\frac{\gamma R T}{M}}

This formula tells you three key things:

Speed increases with temperature — hotter gas means faster molecules, so the disturbance propagates quicker.
Speed decreases with heavier molecules — a gas like helium (small $M$ ) has a much higher speed of sound than air. In helium, your voice sounds squeaky because sound travels faster, changing the resonance in your throat.
The factor $\gamma$ matters — it accounts for the fact that compressions and rarefactions in a sound wave happen so fast that heat doesn't have time to flow. The process is adiabatic, not isothermal.

Why Adiabatic? (The "Why" Behind the Formula)

When a sound wave passes through a gas, the pressure and volume change rapidly — hundreds or thousands of times per second. There's no time for heat to flow from the compressed (hotter) regions to the rarefied (cooler) regions. So the gas behaves as if it's thermally isolated. That's why $\gamma$ appears instead of 1 (which would be the isothermal case).

If you used the isothermal assumption, you'd get $v = \sqrt{RT/M}$ , which is about 20% too low for air. The correct adiabatic formula matches experiments beautifully.

A Quick Numerical Check

At room temperature ( $T = 293\ \text{K}$ ), for air ( $M \approx 0.029\ \text{kg/mol}$ , $\gamma = 1.4$ ):

v = \sqrt{\frac{1.4 \times 8.314 \times 293}{0.029}} \approx \sqrt{117,600} \approx 343\ \text{m/s}

That's about 1235 km/h — the familiar value you've probably heard. …

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The speed of sound in a gas is proportional to the square root of its absolute temperature, $v\propto\sqrt T$ . Tripling the speed requires the absolute temperature to increase ninefold; starting from $0^\circ\text{C}=273\ \text{K}$ , this gives $T=2457\ \text{K}$ , i.e. $\boxed{2184^\circ\text{C}}$ .

The governing relationship

For an ideal gas, the speed of sound is $v = \sqrt{\dfrac{\gamma RT}{M}}$ , where $\gamma$ , $R$ , and $M$ are constants for a given gas (air). So for the same gas, $v$ depends on temperature only through $\sqrt T$ : $v \propto \sqrt T$ . Crucially, $T$ here must be the absolute temperature (Kelvin).

Setting up the ratio

Let $v_0$ be the speed of sound at $T_0=0^\circ\text{C}$ , and $v$ the speed at the unknown temperature $T$ , with $v=3v_0$ . The constants cancel in the ratio:

\frac{v}{v_0} = \sqrt{\frac{T}{T_0}} \quad\Rightarrow\quad 3 = \sqrt{\frac{T}{T_0}}

Solving for T

Squaring both sides: $9 = \dfrac{T}{T_0} \Rightarrow T = 9T_0$ .

Using $T_0 = 0^\circ\text{C} = 273\ \text{K}$ (the standard exam convention):

$T = 9\times273 = 2457\ \text{K}$ …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2026Set ANNUAL1 mark
$Q. Fill in the blank: When a sound wave enters into water from air, then physical quantity ________ remains unchanged.$
›Reveal solutionSolution

When a sound wave travels from air into water, its frequency stays the same; only its speed and wavelength change.

The frequency of a wave is determined by the source producing it — every particle in the new medium is forced to oscillate at the same rate as the source, so the frequency cannot change just because the medium changes. However, the wave's speed does change (sound travels much faster in water, ~1480 m/s, than in air, ~340 m/s, because water is denser and less compressible in the relevan …

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CBSE 2026Set ANNUAL1 markMCQ
$Q. The speed of Sound wave in gas :$
(a) Does not depend upon density of the gas.
(b) Does not depend upon change in pressure.
(c) Does not depend upon temperature.
(d) None of these.
›Reveal solutionSolution

By the Newton-Laplace formula v = √(γP/ρ), the ratio P/ρ stays constant at a given temperature (since ρ is proportional to P for a fixed amount of gas), so the speed of sound in a gas does not change with pressure alone.

The speed of sound in a gas is given by the Newton-Laplace formula:

v = √(γP/ρ)

where P is pressure, ρ is density, and γ = Cp/Cv.

From the ideal gas equation, PV = nRT, so PM = ρRT (M = molar mass), giving:

P/ρ = RT/M

…

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CBSE 2026Set ANNUAL1 markMCQ
$Q. On increasing the temperature of air by 1°C, velocity of sound in it increases by$
(a) 0.331 m/s
(b) 0.273 m/s
(c) 0.232 m/s
(d) 0.607 m/s
›Reveal solutionSolution

Speed of sound in air increases by about 0.61 m/s per degree C. Answer (D).

The speed of sound in air varies with temperature approximately as v = 331 + 0.61 t (m/s), with t in degrees C.

…

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CBSE 2025Set ANNUAL1 markMCQ
$Q. Speed of sound in gas is independent from (A) temperature (B) pressure (C) humidity (D) the effect of all these three$
›Reveal solutionSolution

Speed of sound in a gas is independent of pressure, although it does depend on temperature and (weakly) on humidity.

Speed of sound in a gas: $v = \sqrt{\dfrac{\gamma P}{\rho}}$

For an ideal gas at fixed temperature, density $\rho$ is directly proportional to pressure $P$ (from $PV=nRT \Rightarrow \rho = PM/RT$ ), so the ratio $P/\rho$ stays constant even if $P$ changes — meaning $v$ does not change with pressure at constant temperature.

…

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CBSE 2025Set ANNUAL1 markMCQ
$Q. The ratio of speed of sound in air at 4 and 1 atmospheric pressure will be (A) 1 : 1 (B) 1 : 4 (C) 4 : 1 (D) 3 : 1$
›Reveal solutionSolution

The speed of sound in air is the same at 4 atm and 1 atm — ratio 1:1.

Speed of sound: $v = \sqrt{\gamma P/\rho}$ . For a fixed mass of gas at a given (unchanged) temperature, density is proportional to pressure ( $\rho \propto P$ ), so $P/\rho$ is a constant independent of the actual pressure value. Hence $v$ is unaffected by a chang …

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CBSE 2025Set ANNUAL1 markMCQ
$Q. Assertion (A) : The flash of lightning is seen before the sound of thunder is heard. Reason (R) : The speed of sound is greater than the speed of light.$
(a) (A) and (R) are both true and (R) is the correct explanation of (A).
(b) (A) and (R) are both true but (R) is not the correct explanation of (A).
(c) (A) is false but (R) is true
(d) (A) is true, but (R) is false
›Reveal solutionSolution

The Assertion is true, but the Reason is false — light is much faster than sound, not slower.

Assertion: The flash of lightning is seen before the sound of thunder is heard — this is TRUE. Both the flash and the thunder are produced at essentially the same instant (the electrical discharge), but they reach an observer at very different times because they travel at very different speeds.

…

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CBSE 2024Set ANNUAL1 markMCQ
$Q. On increasing the temperature, the speed of sound in gaseous medium (A) decreases (B) increases (C) does not change (D) increases or decreases depending on the nature of gas$
›Reveal solutionSolution

Speed of sound in a gas increases with temperature, since $v\propto\sqrt T$ .

The speed of sound in a gas is given by Laplace's formula, $v=\sqrt{\dfrac{\gamma P}{\rho}}=\sqrt{\dfrac{\gamma RT}{M}}$ (using the ideal gas law), where $T$ is the absolute temperature. Since …

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CBSE 2024Set SET-AP55001 markMCQ
$Q. In which of these will the speed of sound be maximum?$
(a) Water
(b) Iron
(c) Vacuum
(d) Air
›Reveal solutionSolution

Speed of sound depends on the medium's elasticity and density (v = √(E/ρ) for solids); solids like iron have by far the highest speed of sound among solids, liquids and gases.

Sound needs a material medium to travel (it's a mechanical wave), so it cannot travel through a vacuum at all — this immediately rules out option (c).

…

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CBSE 2024Set SET-NDP60001 mark
$Q. The sound has maximum speed in ............. (Solid/ gas).$
›Reveal solutionSolution

Sound travels fastest in solids because their elasticity (bulk/rigidity modulus) is far higher than that of liquids or gases.

The speed of sound in a medium is given generally by $v = \sqrt{E/\rho}$ , where $E$ is the relevant modulus of elasticity of the medium and $\rho$ is its density. Although solids are also denser than gases, their elastic modulus is enormously larger — solids strongly resist both compression and shear — and this dominates the ratio $E/\rho$ . …

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CBSE 2022Set TERM11 markMCQ
$Q. Speed of sound wave in air$
(1) is independent of temperature
(2) increases with increase in pressure
(3) increases with the increase in humidity
(4) decreases with the increase in humidity
›Reveal solutionSolution

Speed of sound in a gas is v = sqrt(gamma RT/M). Humid air has a lower effective molar mass M than dry air (water vapour molecules are lighter than N2/O2), so sound speed INCREASES with humidity; it is essentially independent of pressure at constant temperature, and strongly dependent on (increases with) temperature.

The speed of sound in an ideal gas is given by:

v = sqrt(gamma R T / M)

where gamma is the adiabatic index, R the gas constant, T the absolute temperature, and M the molar mass of the gas.

Checking each option:

(1) 'independent of temperature' -- FALSE; v is proportional to sqrt(T), so sound speed clearly depends on (increases with) temperature.

(2) 'increases with increase in pressure' -- FALSE; at constant temperature, increasing pressure also proportionally increases density, and these effects cancel in the formula v = sqrt(gamma P/rho) -- sound speed is essentially independent of pressure at constant T. …

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CBSE 2020Set ANNUAL1 markMCQ
$Q. Speed of sound wave in air:$
(a) is independent of temperature
(b) increases with increase in pressure
(c) increases with the increase in humidity
(d) decreases with the increase in humidity
›Reveal solutionSolution
Speed of sound in air depends strongly on temperature (increasing with it) and also increases with humidity, because moist air has a lower density than dry air.

The speed of sound in a gas is given by v = sqrt(gamma P / rho) (or equivalently, for an ideal gas, v = sqrt(gamma R T / M)).

Temperature: v increases with increasing temperature (since v is proportional to sqrt(T)), so option (a) 'independent of temperature' is wrong.

Pressure: at constant temperature, P and rho both change proportionally for a given amount of gas (P/rho is roughly constant for an ideal gas at fixed T), so speed of sound is essentially independent of pressure alone, ruling out option (b). …
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Q.At what temperatures (in ∘^{\circ}∘C) will the speed of sound in air be 3 times its value at 0∘0^{\circ}0∘C?

The governing relationship

Setting up the ratio

Solving for T

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Q.At what temperatures (in $^{\circ}$ C) will the speed of sound in air be 3 times its value at $0^{\circ}$ C?

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