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.)
Answers
Answer:
1.133 kPa is the average pressure exerted by the molecules on the walls of the container.
Explanation:
Side of the cubic box = s = 20.0 cm
Volume of the box ,V=
Root mean square speed of the of helium molecule : 200m/s
The formula used for root mean square speed is:
where,
= root mean square speed
k = Boltzmann’s constant =
T = temperature = 370 K
M = mass helium =
= Avogadro’s number =
Moles of helium gas = n
Number of helium molecules = N =
N =
Ideal gas equation:
PV = nRT
Substitution of values of T and n from above :
(1 Pa = 0.001 kPa)
1.133 kPa is the average pressure exerted by the molecules on the walls of the container.
Final answer:
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.
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.
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Explanation:
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