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The pressure p and volume v of an expanding gas are related by the formula pvb = c, where b and c are constants. (this holds in adiabatic expansion, with or without heat gain or loss.) find b if p = 33 kpa, dp dt = 9 kpa/min, v = 80 cm3 and dv dt = 23 cm3/min.

Respuesta :

The given expression is
[tex]pV^{b} = c[/tex]
where b and c are constants.

Take logs of both sides.
ln p + b ln V = ln c

Take partial derivatives.
[tex] \frac{1}{p} \frac{\partial p}{\partial t} + \frac{b}{V} \frac{\partial V}{\partial t} =0[/tex]

Given:
p = 33 kPa = 33 x 10³ N/m²
V = 80 cm³ = 80 x 10⁻⁶ m³
dp/dt = 9 kPa/min = (9 x 10³ N/m²)/(60 s) = 150 N/m²-s)
dV/dt = 23 cm³/min = (23 x 10⁻⁶  m³)/(60 s) = 3.8333 x 10⁻⁷ m³/s

Therefore
1/(33 x 10³) * 150 + b/(80 x 10⁻⁶) * 3.8333 x 10⁻⁷ = 0
4.5455 x 10⁻³ + 4.7917 x 10⁻³ b = 0
b = - 0.9486

ln c = ln(33 x 10³) - 0.9486 * ln(80 x 10⁻⁶) = 19.3529
c = exp(19.3529) = 2.54x 10⁸

Answer:
b = 2.54 x 10⁸