A sealed container of volume V =0 .10m^3 holds a sample of N =3 .0×10^24 atoms of helium gas in equilibrium. The distribution of speeds of the helium atoms shows a peak at 1.1×10^3 ms^−1.
(i) Calculate the temperature and pressure of the helium gas.
(ii) What is the average kinetic energy of the helium atoms?
(iii) What is the energy corresponding to the maximum in the energy distribution
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I have the temperature as 194K. for the Pressure I need to use P=nKT I assume but im a little lost on the value of n. I have the number of helium atoms as 3.0×10^24 so do I times this by the amu and then by Avocado(joke) constant to get the n value of moles?
Any help would be greatly accepted! Thank you.
number of mols, n = 3*10^24/6*10^23 = 5
So if I know the average kinetic energy of one mole of helium atoms. Can I just times this by five to obtain the answer to ii) ? this seems deceptively simple.
yes, if you know the avg for one mole, multiply it by 5.
To answer these questions, we need to use the ideal gas law, which states that PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant (8.31 J/(mol·K)), and T is the temperature in Kelvin. Let's go through each question step by step:
(i) To calculate the temperature, we know the volume (V = 0.10 m^3) and the number of atoms (N = 3.0×10^24). However, we need to convert the number of atoms into moles (n). Since we know Avogadro's number (6.022×10^23 atoms/mol), we can divide the number of atoms by Avogadro's number to get the number of moles:
n = N / Avogadro's number
n = 3.0×10^24 / 6.022×10^23
n ≈ 4.98 moles (approximately)
Now, we can rearrange the ideal gas law to solve for temperature:
T = (PV) / (nR)
T = (P * 0.10 m^3) / (4.98 mol * 8.31 J/(mol·K))
Substituting the known values, we have:
T ≈ (P * 0.10) / (4.98 * 8.31) (in Kelvin)
(ii) The average kinetic energy of the helium atoms can be calculated using the kinetic energy formula:
Average kinetic energy = (3/2) * k * T
Where k is the Boltzmann constant (1.38×10^-23 J/K) and T is the temperature in Kelvin. Let's substitute the value of T we found earlier into the equation:
Average kinetic energy = (3/2) * (1.38×10^-23 J/K) * T
(iii) The energy corresponding to the maximum in the energy distribution is also called the most probable kinetic energy. The most probable kinetic energy is given by:
Most probable kinetic energy = (1/2) * k * T
Substituting the value of T, we can find the energy corresponding to the maximum in the energy distribution.
Note: Remember to use consistent units throughout the calculations, converting as necessary.
I hope this helps! Let me know if you have any further questions.