A sealed container of volume 0.10 m3 holds a sample of 3.0 x 10^24 atoms of helium gas in equilibrium. The distribution of speeds of helium atoms shows a peak at 1100 ms-1.
Caluculate the temperature and pressure of the helium gas. (helium atom 4.0 amu)
How do I calculate the mass of helium? I got 2.0 x 10^-2 kg from (sample size/avogrado's constant) x 4.0 x 10^-3 kg.
Is this correct as when I put it into
T = (v^2 x m)/2k
I get a huge temp in the 8^30 K region which cannot be right!
What am I doing wrong here?
When I get the temp I can get the pressure from the ideal gas equation.
You can get the temperature from the velocity distribution curve. The average kinetic energy is (1/2) m V^2 = (3/2) k T
where V^2 is the average V^2 of the atoms.
This tells you that the average V^2 is
V^2 = 3 k T/m, so
T = mV^2/(3 k)
k is the Boltzmann constant 1.3804*10^-23 Joule/K
m is the mass of a helium atom, which is 4.0 g/6.023*10^23 = 6.64*10^-27 kg.
The peak of the velocity distribution is not the same as the square root of the averge V^2, but is is pretty close. You can worry about the difference later. Using sqrt(Vpeak^2) for average V^2, I get
T = 6.64*10^-27 kg*(1100^2)m^2/s^2/3*(1.38*10^-23)J/K = 194 K
Now that you know T and the number density n (atoms/m^3), the equation
P = n k T
can be used for the pressure.
k is Boltzmann's constant. You could also use P = N R T, where R is the gas constant and N is the number of MOLES per m^3 and r is the gas constant.
Now for a fine point. In the Maxwellian velocity distribution of atoms, which is what you have, the peak velocity Vp differs from Vrms.
Vp = sqrt(2/3)* Vrms = 0.816 Vrms
See http://hyperphysics.phy-astr.gsu.edu/Hbase/kinetic/kintem.html
for an explanation.
To get the temperature, I should have used Vp instead of Vrms. This has the effect of lowering the temperature by a factor 2/3.
Thanks for the help drwls.
I have now also found the average kinectic energy of the helium atoms.
But what is the position of the maximum in the energy distribution? Is this the peak energy that I need to find?
Well, it seems like you've run into a bit of a pickle there! Let's try to sort it out.
First off, your calculation for the mass of helium is incorrect. The atomic mass of helium is actually 4.0 atomic mass units (amu), which is equivalent to 6.64 x 10^-27 kilograms. So, instead of multiplying by 4.0 x 10^-3 kg, you should multiply by 6.64 x 10^-27 kg.
Once you have the correct mass, you can plug it into the equation:
T = (v^2 x m) / (2k)
where:
T is the temperature,
v is the peak velocity (1100 m/s in this case),
m is the mass of the helium atom,
and k is the Boltzmann constant (1.38 x 10^-23 J/K).
By plugging in the correct values, this should give you a more sensible temperature. Remember, we're dealing with very small masses here, so the temperature shouldn't be in the 8^30 K region!
Once you've calculated the temperature, as you mentioned, you can then use the ideal gas equation to find the pressure. Just make sure to use the correct units!
Keep in mind that my calculations are only as reliable as a clown balancing on a unicycle, so always double-check your work and consult with your teacher or textbook if in doubt. Good luck!
To calculate the mass of helium, you need to multiply the sample size (3.0 x 10^24 atoms) by the mass of each helium atom. The mass of a helium atom is given as 4.0 amu, which stands for atomic mass units.
However, we need to convert amu to kg before proceeding with the calculation. The conversion factor is 1 amu = 1.66 x 10^-27 kg.
So, the mass of helium can be calculated as follows:
Mass of helium = (sample size) x (mass of each helium atom) x (conversion factor)
= (3.0 x 10^24 atoms) x (4.0 amu) x (1.66 x 10^-27 kg/amu)
≈ 1 x 10^-2 kg
Now, let's plug this value into the equation you mentioned:
T = (v^2 x m) / (2k)
Given that the distribution of speeds of helium atoms shows a peak at 1100 m/s, and the value of m is now 1 x 10^-2 kg, we'll use the appropriate SI units:
v = 1100 m/s
m = 1 x 10^-2 kg
k = Boltzmann constant = 1.38 x 10^-23 J/K
Now, let's calculate the temperature (T):
T = (v^2 x m) / (2k)
= ((1100 m/s)^2 x (1 x 10^-2 kg)) / (2 x (1.38 x 10^-23 J/K))
≈ 5.02 x 10^5 K
So, the temperature of the helium gas is approximately 5.02 x 10^5 K.
Once you have the temperature, you can use the ideal gas equation to calculate the pressure (P). The ideal gas equation is given as:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles
R = ideal gas constant = 8.31 J/(mol·K)
T = temperature
To calculate the pressure, we need to convert the number of atoms (sample size) to the number of moles. The conversion factor is Avogadro's constant, which is approximately 6.02 x 10^23 mol^-1.
Number of moles (n) = (sample size) / (Avogadro's constant)
= (3.0 x 10^24 atoms) / (6.02 x 10^23 mol^-1)
≈ 4.98 mol
Now, substitute the values into the ideal gas equation:
(P) x (0.10 m^3) = (4.98 mol) x (8.31 J/(mol·K)) x (5.02 x 10^5 K)
P ≈ 2.07 x 10^7 Pa
Therefore, the pressure of the helium gas is approximately 2.07 x 10^7 Pa.
To calculate the mass of helium, you correctly used the formula (sample size/Avogadro's constant) multiplied by the atomic mass of helium. In this case, the sample size is given as 3.0 x 10^24 atoms and the atomic mass of helium is 4.0 atomic mass units (amu).
The atomic mass unit is defined as 1/12th the mass of a carbon-12 atom, so it is approximately equal to 1.66 x 10^-27 kg. Therefore, the mass of helium can be calculated as follows:
Mass of helium = (3.0 x 10^24 atoms / 6.022 x 10^23 atoms/mol) x (4.0 amu x 1.66 x 10^-27 kg/amu)
= 2.0 x 10^-2 kg
It seems like you have correctly calculated the mass of helium. However, the issue lies in the equation you used to calculate the temperature. The equation you mentioned, T = (v^2 x m) / (2k), is not the correct formula to calculate temperature.
The correct formula to calculate temperature in this case is:
T = (m x v^2) / (3k),
where T is the temperature, m is the mass of the gas, v is the most probable speed (peak speed), and k is the Boltzmann constant (approximately equal to 1.38 x 10^-23 J/K).
Using this formula, we can calculate the temperature of helium:
T = (2.0 x 10^-2 kg x (1100 m/s)^2) / (3 x 1.38 x 10^-23 J/K)
≈ 3.80 x 10^6 K
So, the temperature of the helium gas is approximately 3.80 x 10^6 Kelvin.
Once you have the temperature, you can calculate the pressure using the ideal gas equation, which 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.
In this case, since the container is sealed and in equilibrium, the number of moles (n) can be calculated as follows:
n = (sample size / Avogadro's constant) ≈ (3.0 x 10^24 atoms / 6.022 x 10^23 atoms/mol) ≈ 5 moles.
Given the volume of the container as 0.10 m^3, the value of the ideal gas constant (R) is approximately 8.31 J/(mol·K), you can calculate the pressure (P) as follows:
P = (nRT) / V
≈ (5 moles x 8.31 J/(mol·K) x 3.80 x 10^6 K) / 0.10 m^3
≈ 1.57 x 10^9 Pa
Therefore, the pressure of the helium gas is approximately 1.57 x 10^9 Pascal.