Gas Laws

Using the Formula for Root Mean Square (r.m.s.) Velocity: The r.m.s. velocity $v_%7Brms%7D$ for a gas is given by:

$v_%7Brms%7D%20%3D%20%5Csqrt%7B%5Cfrac%7B3RT%7D%7BM%7D%7D$

where $R$ is the gas constant, $T$ is the temperature, and $M$ is the molar mass of the gas.

Set up the Equation for Hydrogen and Oxygen: To find the temperature at which the r.m.s. velocity of hydrogen equals that of oxygen at 47°C, we set:

$%5Csqrt%7B%5Cfrac%7B3RT_%7BH_2%7D%7D%7BM_%7BH_2%7D%7D%7D%20%3D%20%5Csqrt%7B%5Cfrac%7B3RT_%7BO_2%7D%7D%7BM_%7BO_2%7D%7D%7D$

Isolate $T_%7BH_2%7D$ : Square both sides to remove the square root:

$%5Cfrac%7B3RT_%7BH_2%7D%7D%7BM_%7BH_2%7D%7D%20%3D%20%5Cfrac%7B3RT_%7BO_2%7D%7D%7BM_%7BO_2%7D%7D$

Simplify by canceling $3R$ on both sides:

$T_%7BH_2%7D%20%3D%20T_%7BO_2%7D%20%5Ctimes%20%5Cfrac%7BM_%7BH_2%7D%7D%7BM_%7BO_2%7D%7D$

Substitute Values for Molar Mass and Temperature: Given $T_%7BO_2%7D%20%3D%2047%C2%B0C%20%3D%20320%20%5C%2C%20K$ ,

$T_%7BH_2%7D%20%3D%20320%20%5Ctimes%20%5Cfrac%7B2%7D%7B32%7D%20%3D%2020%20%5C%2C%20K$

This problem involves the application of the Ideal Gas Law,

PV%20%3D%20nRT

, where

P

is pressure,

V

is volume,

n

is the number of moles,

R

is the universal gas constant, and

T

is temperature. Initially, we have two vessels of volume

V

, each at pressure

P_i%20%3D%2090%5Ctext%7B%20kPa%7D

and temperature

T_i%20%3D%20400%5Ctext%7B%20K%7D

. The total number of moles of air in the system is:
$

n_%7Btotal%7D%20%3D%20n_1%20%2B%20n_2%20%3D%20%5Cfrac%7BP_i%20V%7D%7BR%20T_i%7D%20%2B%20%5Cfrac%7BP_i%20V%7D%7BR%20T_i%7D%20%3D%20%5Cfrac%7B2%20P_i%20V%7D%7BR%20T_i%7D

%3Cbr%3ESubstituting%20the%20initial%20values%3A%3Cbr%3E

n_%7Btotal%7D%20%3D%20%5Cfrac%7B2%20%5Ctimes%2090%20%5Ctimes%20V%7D%7BR%20%5Ctimes%20400%7D%20%3D%20%5Cfrac%7B180%20V%7D%7B400%20R%7D%20%3D%20%5Cfrac%7B9%20V%7D%7B20%20R%7D

%3Cbr%3EIn%20the%20final%20state%2C%20the%20temperature%20of%20vessel%201%20remains

T_1 = 400\text{ K}

and%20the%20temperature%20of%20vessel%202%20is%20raised%20to

T_2 = 500\text{ K}

.%20Let%20the%20final%20common%20pressure%20in%20both%20vessels%20be

P_f

.%20Since%20the%20system%20is%20closed%2C%20the%20total%20number%20of%20moles%20remains%20constant%3A%3Cbr%3E

n_%7Btotal%7D%20%3D%20%5Cfrac%7BP_f%20V%7D%7BR%20T_1%7D%20%2B%20%5Cfrac%7BP_f%20V%7D%7BR%20T_2%7D%20%3D%20%5Cfrac%7BP_f%20V%7D%7BR%7D%20%5Cleft(%20%5Cfrac%7B1%7D%7B400%7D%20%2B%20%5Cfrac%7B1%7D%7B500%7D%20%5Cright)

%3Cbr%3EEquating%20the%20expressions%20for

n_{total}

from%20the%20initial%20and%20final%20states%3A%3Cbr%3E

%5Cfrac%7B9%20V%7D%7B20%20R%7D%20%3D%20%5Cfrac%7BP_f%20V%7D%7BR%7D%20%5Cleft(%20%5Cfrac%7B5%20%2B%204%7D%7B2000%7D%20%5Cright)%20%3D%20%5Cfrac%7BP_f%20V%7D%7BR%7D%20%5Ccdot%20%5Cfrac%7B9%7D%7B2000%7D

%3Cbr%3ESimplifying%20the%20equation%20by%20canceling

V/R

and%20the%20factor%20of

%3A%3Cbr%3E

%5Cfrac%7B1%7D%7B20%7D%20%3D%20%5Cfrac%7BP_f%7D%7B2000%7D

%3Cbr%3E

P_f%20%3D%20%5Cfrac%7B2000%7D%7B20%7D%20%3D%20100%5Ctext%7B%20kPa%7D

%3Cbr%3EThe%20final%20pressure%20in%20the%20vessels%20is

100\text{ kPa}$.

Chapter Questions
6 MCQs