Speed of sound wave in air (a) is independent of temperature. (b) increases with pressure. (c) increases with increase in humidity. (d) decreases with increase in humidity.

Speed of sound wave in air (a) is independent of temperature. (b) increases with pressure. (c) increases with

NCERT Exemplar · Q3

Q. Speed of sound wave in air

(a) is independent of temperature.

(b) increases with pressure.

(d) decreases with increase in humidity.

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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.

Tip

A handy rule: For dry air at 0°C, the speed of sound is about 331 m/s. For every 1°C rise, it increases by roughly 0.6 m/s. So at 20°C, it's about 331 + 12 = 343 m/s.

Common Mistake to Avoid

Watch out

Do not use the formula $v = \sqrt{P/\rho}$ (which comes from the bulk modulus) without remembering that for a gas, the bulk modulus is $\gamma P$ , not $P$ . The correct form is $v = \sqrt{\gamma P / \rho}$ , which is equivalent to the formula above because $P = \rho R T / M$ for an ideal gas.

Why This Matters for Exams

You'll often be asked to compare speeds in different gases (e.g., He vs. air, or H₂ vs. O₂) at the same temperature. Use $v \propto 1/\sqrt{M}$ .
Temperature dependence is a favourite: $v \propto \sqrt{T}$ .
The adiabatic nature is a conceptual question — be ready to explain why $\gamma$ appears.

Sound in gases is a beautiful blend of thermodynamics and wave physics. The formula isn't just a plug-and-chug tool — it tells a story about how molecules jostle and pass along energy.

This is exactly the kind of concept that turns up under searches like "Speed of Sound in Gases class 11 physics syllabus" or "Speed of Sound in Gases solved examples" — and it belongs squarely in the Class 11 Physics NCERT/CBSE curriculum. Beyond board exams, it's a dependable scoring topic in JEE Main, NEET and state engineering/medical entrance exams once the core logic clicks.

The speed of sound in a gas depends on the medium's density and elasticity. For air, increasing humidity replaces heavier nitrogen/oxygen molecules with lighter water vapour, lowering density and thus increasing the speed of sound. The correct option is (C).

Concept First: Why Sound Speed Changes in Gases

Sound travels as a mechanical wave through a medium. In a gas, the speed is governed by Newton-Laplace formula:

v = \sqrt{\frac{\gamma P}{\rho}}

where $\gamma$ is the adiabatic index (ratio of specific heats), $P$ is pressure, and $\rho$ is density. This formula tells us that the speed depends on two things: the stiffness (elasticity) of the medium and its inertia (density). For an ideal gas, we can rewrite this using the ideal gas law $PV = nRT$ to get:

v = \sqrt{\frac{\gamma RT}{M}}

where $M$ is the molar mass of the gas. This form is crucial — it shows that for a given gas, speed depends only on temperature, not on pressure (since $P$ and $\rho$ change together in a way that cancels out).

Now let's examine each option carefully.

Option (A): "is independent of temperature" — This is false. From $v \propto \sqrt{T}$ , we see that speed increases with the square root of absolute temperature. Warmer air conducts sound faster.
Option (B): "increases with pressure" — This is a classic trap. At constant temperature, if you increase pressure, density increases proportionally (Boyle's law), so $P/\rho$ remains constant. The speed is independent of pressure at a fixed temperature. So this is false.
Options (C) and (D): Effect of humidity — Here's where the real physics lies. Humid air is a mixture of dry air (mostly $N_2$ and $O_2$ , molar mass ~29 g/mol) and water vapour ( $H_2O$ , molar mass 18 g/mol). Water molecules are lighter.

Tip

Think of it this way: when you add water vapour, you're replacing some heavier molecules with lighter ones. The mixture's average molar mass decreases. Since $v \propto 1/\sqrt{M}$ (at constant temperature), the speed increases.

Let's verify this mathematically. For a mixture of dry air and water vapour, the effective molar mass $M_{eff}$ is less than that of dry air. Since $v = \sqrt{\gamma RT / M}$ , a smaller $M$ gives a larger $v$ . So humidity increases the speed of sound.

Watch out

A common mistake is to think that humidity makes air "heavier" because water vapour feels damp. But water vapour is actually less dense than dry air at the same temperature and pressure — that's why humid air rises (think of clouds).

Therefore, option (C) is correct: speed increases with increase in humidity.

✓Final answer

The correct option is (C) : the speed of sound in air increases with an increase in humidity.

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Q.Speed of sound wave in air

Concept First: Why Sound Speed Changes in Gases

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