Question

A cubic box with sides of 20.0 cm contains 2.00 × 1023 molecules of helium with a root-mean-square speed (thermal speed) of 200 m/s. The mass of a helium molecule is 3.40 × 10-27 kg. What is the average pressure exerted by the molecules on the walls of the container? (The Boltzmann constant is 1.38 × 10-23 J/K and the ideal gas constant is R = 8.314 J/mol•K .) (12 pts.)

Answer 1

The question asks for the average pressure exerted by helium gas molecules on the walls of a cubic container. Using the equation PV = Nmv^2, we can calculate pressure by substituting the given values for volume, number of molecules, mass of one molecule, and root-mean-square speed.

Step-by-step explanation:

The question is asking to calculate the average pressure exerted by helium gas molecules on the walls of a cubic container. The important formula relating pressure (P), volume (V), number of molecules (N), mass of a molecule (m), and the square of the rms speed (v2) of the molecules in a gas is:

PV = Nmv2,

First, we need to determine the volume of the container, which is the cube of one side, so V = (20 cm)3 = (0.2 m)3. Inserting the given values into the equation and solving for P gives us the desired answer. Recall that the rms speed is given, so no temperature calculations are needed.

Therefore, using all given data points:

Volume (V) = (0.2 m)3

Number of molecules (N) = 2.00 × 1023

Mass of one helium molecule (m) = 3.40 × 10-27 kg

Root-mean-square speed (vrms) = 200 m/s

By substituting these values, we can find the pressure exerted by the gas. This represents an application of kinetic theory of gases which assumes the behavior of an ideal gas.

Answer 2

1.133 kPa is the average pressure exerted by the molecules on the walls of the container.

Step-by-step explanation:

Side of the cubic box = s = 20.0 cm

Volume of the box ,V=
`s^3`

`V=(20.0 cm)^3=8000 cm^3=8* 10^(-3) m^3`

Root mean square speed of the of helium molecule : 200m/s

The formula used for root mean square speed is:

`\mu=\sqrt{(3kN_AT)/(M)}`

where,

= root mean square speed

k = Boltzmann’s constant =
`1.38* 10^(-23)J/K`

T = temperature = 370 K

M = mass helium =
`3.40* 10^(-27)kg/mole`

`N_A` = Avogadro’s number =
`6.022* 10^(23)mol^(-1)`

`T=(\mu _(rms)^2* M)/(3kN_A)`

Moles of helium gas = n

Number of helium molecules = N =
`2.00* 10^(23)`

N =
`N_A* n`

Ideal gas equation:

PV = nRT

Substitution of values of T and n from above :

`PV=(N)/(N_A)* R* (\mu _(rms)^2* M)/(3kN_A)`

`PV=(N* R* \mu ^2* M)/(3k* (N_A)^2)`

`R=k* N_A`

`PV=(N* \mu ^2* M)/(3)`

`P=(2.00* 10^(23)* (200 m/s)^2* 3.40* 10^(-27) kg/mol)/(3* 8* 10^(-3) m^3)`

`P=1133.33 Pa =1.133 kPa`

(1 Pa = 0.001 kPa)

1.133 kPa is the average pressure exerted by the molecules on the walls of the container.