A container holds 1.3 mol of gas. The total average kinetic energy of the gas molecules in the container is equal to the kinetic energy of a 6.7x10-3 kg bullet with a speed of 830 m/s. What is the Kelvin temperature of the gas?
To find the Kelvin temperature of the gas, we need to use the equation of kinetic energy:
KE = (1/2) * m * v^2
Where KE is the kinetic energy, m is the mass, and v is the velocity.
Given that the kinetic energy of the gas is equal to the kinetic energy of a bullet, we can set up the equation:
(1/2) * m_gas * v_gas^2 = (1/2) * m_bullet * v_bullet^2
We are given the mass of the bullet (m_bullet = 6.7x10^(-3) kg) and the velocity of the bullet (v_bullet = 830 m/s). Therefore, we need to find the mass and velocity of the gas to solve for the temperature.
To find the mass of the gas, we need to convert the number of moles to kilograms. The molar mass of the gas can be used for this conversion.
Given that the container holds 1.3 mol of gas, assume the molar mass of the gas is M gas (in g/mol).
Mass of the gas (m_gas) = number of moles of gas (n) * molar mass of the gas (M_gas)
To find the molar mass of the gas, we need more information. Assuming it is a diatomic gas, we can use the average molar mass of diatomic nitrogen (N2) as a representative value:
M_gas = 28.0134 g/mol
Substituting this value and the number of moles into the mass equation:
m_gas = 1.3 mol * 28.0134 g/mol
Now, we can use the equation above to find the velocity of the gas molecules.
(1/2) * m_gas * v_gas^2 = (1/2) * m_bullet * v_bullet^2
Solving for v_gas:
v_gas = sqrt[((m_bullet * v_bullet^2) / m_gas)]
Substitute the known values into this equation and calculate v_gas.
Finally, using the equation for kinetic energy and the value of v_gas, we can find the temperature of the gas using the formula:
KE = (3/2) * R * T
Where KE is the average kinetic energy, R is the gas constant (8.314 J/(mol*K)), and T is the temperature in Kelvin.
Solving for T:
T = (KE / (3/2 * R))
Substitute KE and R with the known values and calculate T.
To find the Kelvin temperature of the gas, we can use the formula for the average kinetic energy of a gas molecule:
KE_avg = (3/2) * k * T
Where KE_avg is the average kinetic energy, k is the Boltzmann constant (1.38 * 10^-23 J/K), and T is the temperature in Kelvin.
First, let's calculate the kinetic energy of the bullet:
KE_bullet = (1/2) * m * v^2
Where KE_bullet is the kinetic energy of the bullet, m is the mass of the bullet (6.7 * 10^-3 kg), and v is the speed of the bullet (830 m/s).
Plugging in the values:
KE_bullet = (1/2) * (6.7 * 10^-3 kg) * (830 m/s)^2
Now, let's find the average kinetic energy of the gas molecules:
KE_avg = KE_bullet
Substituting the values:
(3/2) * k * T = (1/2) * (6.7 * 10^-3 kg) * (830 m/s)^2
Now, solve for T:
T = (1/3) * ((6.7 * 10^-3 kg) * (830 m/s)^2) / k
Calculating the expression on the right side:
T = (1/3) * ((6.7 * 10^-3 kg) * (830 m/s)^2) / (1.38 * 10^-23 J/K)
T ≈ 18914 K
Therefore, the Kelvin temperature of the gas is approximately 18914 K.