Chapter 19, Problem 19.44P

The pressure gauge on a cylinder of gas registers the gauge pressure, which is the difference between the interior pressure and the exterior pressure P₀. Let’s call the gauge pressure P_g. When the cylinder is full, the mass of the gas in it is m_i at a gauge pressure of P_gi. Assuming the temperature of the cylinder remains constant, show that the mass of the gas remaining in the cylinder when the pressure reading is P_gf is given by

$m_f=m_i\left(\frac{P_{gf}+P_0}{P_{gi}+P_0}\right)$