In the Ideal Gas Law formula, there are 3 variables and 2 constants. #P# represents the pressure (in this problem in units of pascals), #V# represents the volume (in this problem in units of #m^3#), and #T# represents the temperature (in this problem in Kelvin). We consider #n# to be the amount of gas present (usually in moles), while #R# is the Ideal Gas Constant, or a fixed value.
This is a "classic" related rates problem. We need to take derivatives of every variable (#P, V, T#) with respect to time #t#. To do so, we perform "traditional" derivatives on each of those terms in the equation, but we ensure that we include a #(dP)/(dt)#, #(dV)/(dt)#, or #(dT)/(dt)# term as necessary.
#PV = nRT#
#Punderbrace((1*(dV)/(dt)))_ "d(V)" + Vunderbrace((1*(dP)/(dt)))_ "d(P)" = nRunderbrace((1*(dT)/(dt)))_ "d(T)"#
#P((dV)/(dt)) + V((dP)/(dt)) = nR((dT)/(dt))#
We have all of the values we need provided to us to substitute into this related rates expression:
#{(P,20 "pascals","Pressure"),((dV)/(dt),-2 m^3/min,"Volume Change"),(V,10 m^3,"Volume"),((dP)/(dt),?,"Pres. Change"),((dT)/(dt),4 K/min,"Temp Change"):}#
#(20)(-2) + (10)((dP)/(dt)) = nR(4)#
#-40 + 10((dP)/(dt)) = 4nR #
#10((dP)/(dt)) = 4nR + 40 #
#(dP)/(dt) = (4nR + 40)/10 = 2/5nR + 4#
A quick "sanity check" verifies that the sign of our answer is what we'd expect. Since P is inversely related to V and directly related to T, if the volume is decreasing, and the temperature is increasing, we'd definitely expect the pressure P to be increasing (positive). Since #n# and #R# are both positive constants, the final expression will be positive.
