Chemistry students will recognize the ideal gas law, given by PV = nRT which rel

Question

Chemistry students will recognize the ideal gas law, given by PV = nRT which relates Pressure (in Pascals), Volume (in Liters), and Temperature (in degrees Kelvin) of n moles or gas where R is the ideal gas constant. Thus, we can view pressure (in Pa), volume (in L), and temperature (in K) as variables, each one dependent on the other two. If pressure of a gas is increasing at a rate of 1.3 Pa/min and temperature is increasing at a rate of 0.9K/min, how fast is the volume changing? If the volume of a gas is decreasing at a rate of 0.2L/min and temperature is increasing at a rate of 0.9 K/min, how fast is the pressure changing? If the pressure of a gas is decreasing at a rate of 0.3 Pa/min and the volume is increasing at a rate of 0.8 L/min, how fast is the temperature changing?

Explanation / Answer

Solution:

PV = n R T
take the derivative of both sides.

P dV + V dP = n R dT
and we can divide the left side by PV and the right side by nRT (since they are equal)

dV / V + dP/P = dT / T

a) dP = 1.3, dT = 0.9
P dv + 1.3 V = n R * 0.9
dv = (0.9nR - 1.3V) / P

b) dV = 0.2, dT = 0.9

P*0.2 + V dP = n R * 0.9

dv = (0.9nR - 0.2P) / V

c) dP = 0.3, dV = 0.8

P dV + V dP = n R dT

P*0.8 + V * 0.3 = n R dT

dT = (0.8P + 0.3V)/ n R

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