A tank holds 135 moles of argon gas (atomic mass = 6.633521 × 10^-26 kg) at a pressure of 101300 Pa and a temperature of 302K. Recall that the number of gas molecules is equal to Avagadros number (6.022 × 10^23) times the number of moles of the gas

1) What is the volume of the tank?
2) How much kinetic energy is contained in this gas?
3) What is the average speed of one of the argon atoms?
4) The escape speed of Earth is 11200 m/s. What temperature would the argon gas need to have so that the average speed of the gas atoms was equal to this escape speed?

To solve these problems, we can use the ideal gas law, which states:

PV = nRT

Where:
P = pressure
V = volume
n = number of moles
R = gas constant
T = temperature

Let's solve each problem step-by-step.

1) What is the volume of the tank?

To find the volume, we can rearrange the ideal gas law equation as follows:

V = (nRT) / P

First, we need to convert the pressure to SI units (Pa). Since 1 atm = 101325 Pa, we have:

P = 101300 Pa

Next, let's substitute the given values into the equation:

V = (135 moles * 6.022 × 10^23 molecules/mole * 8.314 J/(mol·K) * 302 K) / 101300 Pa

Calculating this expression, we find:

V ≈ 0.0316 m^3

Therefore, the volume of the tank is approximately 0.0316 cubic meters.

2) How much kinetic energy is contained in this gas?

The kinetic energy of a gas can be calculated using the following equation:

KE = (3/2) * nRT

Substituting the given values:

KE = (3/2) * (135 moles) * (8.314 J/(mol·K)) * (302 K)

Calculating this expression, we find:

KE ≈ 928,682 J

Therefore, the kinetic energy contained in the gas is approximately 928,682 Joules.

3) What is the average speed of one of the argon atoms?

The average speed of a gas atom can be calculated using the equation:

v = sqrt((3 * k * T) / m)

Where:
v = average speed
k = Boltzmann's constant (1.38 × 10^-23 J/K)
T = temperature
m = mass of one atom

In this case, we need to find the average speed of an argon atom. The atomic mass of argon is given as 6.633521 × 10^-26 kg.

Substituting the values:

v = sqrt((3 * (1.38 × 10^-23 J/K) * (302 K)) / (6.633521 × 10^-26 kg))

Calculating this expression, we find:

v ≈ 474 m/s

Therefore, the average speed of one of the argon atoms is approximately 474 meters per second.

4) What temperature would the argon gas need to have so that the average speed of the gas atoms was equal to the escape speed (11200 m/s)?

To answer this question, we'll first solve the equation for the average speed:

11200 m/s = sqrt((3 * (1.38 × 10^-23 J/K) * T) / (6.633521 × 10^-26 kg))

Squaring both sides of the equation, we get:

(11200 m/s)^2 = (3 * (1.38 × 10^-23 J/K) * T) / (6.633521 × 10^-26 kg)

Simplifying and solving for T, we find:

T ≈ (11200 m/s)^2 * (6.633521 × 10^-26 kg) / (3 * (1.38 × 10^-23 J/K))

Calculating this expression, we find:

T ≈ 102,280 K

Therefore, the temperature that the argon gas needs to have for the average speed of the gas atoms to be equal to the escape speed of Earth (11200 m/s) is approximately 102,280 Kelvin.

To answer these questions, we can use the ideal gas law equation and the kinetic theory of gases. The ideal gas law equation is:

PV = nRT

where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature.

1) To find the volume of the tank, we rearrange the ideal gas law equation to solve for V:

V = (nRT) / P

Substituting the given values:

V = (135 * (6.022 × 10^23) * 8.314 * 302) / 101300

Calculating this expression will give you the volume of the tank.

2) The kinetic energy of a gas can be calculated using the equation:

KE = (3/2) * nRT

Substituting the given values:

KE = (3/2) * (135 * (6.022 × 10^23)) * 8.314 * 302

Calculating this expression will give you the kinetic energy contained in the gas.

3) The average speed of one of the gas atoms can be calculated using the equation:

v_avg = sqrt((3 * k * T) / m)

where v_avg is the average speed, k is the Boltzmann constant (1.38 × 10^-23 J/K), T is the temperature, and m is the mass of one gas atom.

Substituting the given values:

v_avg = sqrt((3 * 1.38 × 10^-23 * 302) / (6.633521 × 10^-26))

Calculating this expression will give you the average speed of one of the argon atoms.

4) To find the temperature at which the average speed of the argon gas atoms is equal to the escape speed of Earth, we can rearrange the equation for the average speed:

T = (m * v_escape^2) / (3 * k)

Substituting the given values:

T = (6.633521 × 10^-26 * (11200^2)) / (3 * 1.38 × 10^-23)

Calculating this expression will give you the temperature needed for the average speed of the gas atoms to be equal to the escape speed of Earth.

Note: In all calculations, ensure that the units are consistent and convert any necessary units.

1) V = nRT/P

2) You mean internal energy? U = 3/2nRT or average translational KE of an individual particle KE = 3/2kT (k is the Boltzman constant)
3)3/2kT = 1/2mv^2. To get mass look up Argon and find its grams per mole and change to kg.
4) SAme equation with v = 11200