The speed of sound v in a gas depends on the gas pressure P and density \rho according to … — JAMB UTME

The speed of sound $v$ in a gas depends on the gas pressure $P$ and density $\rho$ according to the relation $v = k P^a \rho^b$ , where $k$ is a dimensionless constant. What is the numerical value of the exponent $a$ ?

Answer: 0.5

Answer

The numerical value of the exponent

a

0.5

Applying dimensional homogeneity to the relation

v = k P^a \rho^b

yields

[M^0 L^1 T^{-1}] = [M L^{-1} T^{-2}]^a [M L^{-3}]^b

. Equating the powers of time

T

gives

-2a = -1

, which simplifies to

a = 0.5

Step-by-Step Solution

Determine the base dimensions of all physical quantities in the relationship

[v] = M^0 L T^{-1}

[P] = M L^{-1} T^{-2}

, and

[\rho] = M L^{-3}

Physical quantities must be expressed in fundamental dimensions (

M, L, T

) to apply dimensional analysis.

Formulate the dimensional balance equation

M^0 L^1 T^{-1} = M^{a+b} L^{-a-3b} T^{-2a}

By the principle of dimensional homogeneity, the total dimensions on the left side must equal those on the right side.

Equate exponents of

T

to solve for

a

-2a = -1 \implies a = 0.5

Comparing powers of time

T

directly isolates the variable

a

Key Concept

Dimensional Homogeneity and Derivation of Exponents

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