We can start with the ideal gas law:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles of gas, R is the universal gas constant, and T is the absolute temperature. We can rearrange this equation to solve for density (ρ):
ρ = nM/V
where M is the molar mass of the gas and V is the volume. Substituting this into the ideal gas law, we get:
P = (nRT)/V = (ρMV)RT
Now, if we assume that the volume stays constant, we can take the derivative of this equation with respect to temperature:
dP/dT = (ρMV)R
We can also use the chain rule to write this as:
dP/dT = (dP/dρ)*(dρ/dT)
where (dP/dρ) is the partial derivative of pressure with respect to density, and (dρ/dT) is the partial derivative of density with respect to temperature. Rearranging this equation, we get:
(dρ/dT) = (dP/dρ)*(1/((ρMV)R))
Now, we can use the ideal gas law to express (dP/dρ) in terms of pressure and density:
(dP/dρ) = (RT/V)
Substituting this in the previous equation, we get:
(dρ/dT) = (RT/V)*(1/((ρMV)R))
Simplifying this, we get:
(dρ/dT) = (-ρ/VM)*(β)
where β is defined as:
β = (1/V)*(dV/dT)
Now, assuming that the gas is at a constant pressure and volume, we can write:
∆ρ = (dρ/dT)*∆T
Substituting in the previous equation, we get:
∆ρ = (-ρ/VM)*(β)*∆T
Simplifying this, we get:
∆ρ = -ρβ∆T
which is the desired expression.
Show that ∆ρ=-ρβ∆T
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