k, T, Rgas
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"The speed of sound c in an ideal gas be a function only of pressure P and gas density ? . Showing all your work, use dimensional analysis to find the functional relationship between these parameters. Verify that your results are consistent with the equation for speed of sound in an ideal gas, c = √kRgas T .
Given:
The speed of sound c in an ideal gas is depends on pressure P and gas density ρ.
Let us derive the relation between c, P and ρ.
Let us assume that
c=K Px ρy….(1)
K=constant which is dimensionless
The dimension of c= [L1 M0 T-1 ]
The dimension of P= [L-1 M1 T-2 ]
and the dimension of ρ= [L-3 M1 T0 ]
Therefore equation(1) can be dimensionally written as
[L1 M0 T-1 ]=K [L-1 M1 T-2 ]x [L-3 M1 T0 ]y
[L1 M0 T-1 ]=K [L-x Mx T-2x ] [L-3y My T0 ]
Therefore [L1 M0 T-1 ]=K [L-x-3y Mx+y T-2x ]….(2)
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