How do you calculate the temperature of the gas at this new pressure?

Question

A sample of an ideal gas is sealed in a container whose volume cannot change. Initially it has a pressure of 2.5 x 10^5 Pa and its temperature -20 oC.The gas is cooled until the pressure is reduced to 1.0 x 10^5 Pa.

Natalie Anton · Accepted Answer

Since you are talking about volume, pressure, and temperature, we can naturally use the ideal gas law. #\mathbf(PV = nRT)# where: You are told that the volume cannot change, so the volume stays constant. Note your variables: #P_1 = 2.5xx10^5 "Pa"#
#P_2 = 1.0xx10^5 "Pa"#
#T_1 = -20^@ "C" = "253.15 K"#
#T_2 = ?# So, set up your equations using these variables. We know that #V#, #n#, and #R# are the same across both equations because the container is closed (constant #n#) and rigid (constant #V#).  (Obviously, the universal gas constant is constant.) #P_1V = nRT_1#
#P_2V = nRT_2# Therefore, we can divide these equations to solve for #T_2#. #(P_1cancel(V))/(P_2cancel(V)) = (cancel(nR)T_1)/(cancel(nR)T_2)# #(P_1)/(P_2) = (T_1)/(T_2)# #=> color(blue)(T_2) = T_1xx(P_2)/(P_1)# #= ("253.15 K")xx(1.0xx10^5 cancel"Pa")/(2.5xx10^5 cancel"Pa")# #=# #color(blue)("101.26 K")# #=# #color(blue)(-171.89^@ "C")#

Naomi Aronson · Answer

undefined To calculate the temperature of the gas at the new pressure, you can use the ideal gas law equation, which is PV = nRT. Rearrange the equation to solve for temperature: T = (P * V) / (n * R), where P is the new pressure, V is the volume of the gas, n is the number of moles of gas, R is the ideal gas constant (8.314 J/(mol·K)), and T is the temperature in Kelvin. Plug in the values of P, V, n, and R to find the temperature.