Calculate the mean speed, using equation 1.13 and 1.17 of (a) He atoms and (b) CH4 molecules at (i) 79 K, (ii) 315K, and (iii) 1500K.
Vrms= (2(Ek)/m)^1/2
Vrms= (3RT/M)^1/2
M(He)= 4.0*10^-3 kg/mol
M(CH4)= 16.0*10^-3 kg/mol
And your problem is? Just plug in the numbers.
To calculate the mean speed of He atoms and CH4 molecules at different temperatures using equations 1.13 and 1.17, we need the values for temperature (T), gas constant (R), and the molar mass (M) of the particles.
Given:
(i) T1 = 79 K
(ii) T2 = 315 K
(iii) T3 = 1500 K
M(He) = 4.0 * 10^-3 kg/mol
M(CH4) = 16.0 * 10^-3 kg/mol
Let's calculate the mean speed:
(a) Mean speed of He atoms:
(i) Using equation 1.13:
Vrms_1 = √((2 * Ek_1) / m_He)
where Ek_1 is the average kinetic energy at temperature T1.
For He atoms, the average kinetic energy Ek is related to the temperature T by the equation Ek = (3/2) * k * T, where k is Boltzmann's constant (k ≈ 1.38 * 10^-23 J/K).
Ek_1 = (3/2) * k * T1
Substituting the values:
Ek_1 = (3/2) * (1.38 * 10^-23 J/K) * 79 K
Now, let's calculate Vrms_1 using equation 1.13:
Vrms_1 = √((2 * Ek_1) / m_He)
= √((2 * (3/2) * (1.38 * 10^-23 J/K) * 79 K) / (4.0 * 10^-3 kg/mol))
(ii) Using the same approach as above, we can calculate Vrms_2 for He atoms at T2 = 315 K.
(iii) Similarly, let's calculate Vrms_3 for He atoms at T3 = 1500 K.
(b) Mean speed of CH4 molecules:
(i) Using equation 1.17:
Vrms_1 = √((3 * R * T1) / M_CH4)
where R is the gas constant, and M_CH4 is the molar mass of CH4 molecules.
Substituting the values:
Vrms_1 = √((3 * R * T1) / (16.0 * 10^-3 kg/mol))
(ii) Using the same approach as above, we can calculate Vrms_2 for CH4 molecules at T2 = 315 K.
(iii) Similarly, let's calculate Vrms_3 for CH4 molecules at T3 = 1500 K.
Please note that the gas constant R ≈ 8.314 J/(mol·K).
Performing these calculations will give us the mean speeds for He atoms and CH4 molecules at each temperature.
To calculate the mean speed of He atoms and CH4 molecules at different temperatures, we will use Equation 1.13 and Equation 1.17.
For He atoms (a):
(i) Temperature = 79 K
(ii) Temperature = 315 K
(iii) Temperature = 1500 K
For CH4 molecules (b):
(i) Temperature = 79 K
(ii) Temperature = 315 K
(iii) Temperature = 1500 K
Let's calculate the mean speed using Equation 1.13 and Equation 1.17:
Using Equation 1.13:
Vrms = sqrt(2*(Ek)/m)
For He atoms (a):
(i) Vrms = sqrt(2*(3*R*(79 K))/(M(He)))
(ii) Vrms = sqrt(2*(3*R*(315 K))/(M(He)))
(iii) Vrms = sqrt(2*(3*R*(1500 K))/(M(He)))
For CH4 molecules (b):
(i) Vrms = sqrt(2*(3*R*(79 K))/(M(CH4)))
(ii) Vrms = sqrt(2*(3*R*(315 K))/(M(CH4)))
(iii) Vrms = sqrt(2*(3*R*(1500 K))/(M(CH4)))
Using Equation 1.17:
Vrms = sqrt((3*R*T)/(M))
For He atoms (a):
(i) Vrms = sqrt((3*R*(79 K))/(M(He)))
(ii) Vrms = sqrt((3*R*(315 K))/(M(He)))
(iii) Vrms = sqrt((3*R*(1500 K))/(M(He)))
For CH4 molecules (b):
(i) Vrms = sqrt((3*R*(79 K))/(M(CH4)))
(ii) Vrms = sqrt((3*R*(315 K))/(M(CH4)))
(iii) Vrms = sqrt((3*R*(1500 K))/(M(CH4)))
To calculate the mean speed, substitute the values of R (ideal gas constant), the respective temperature, and the molecular weight into the equations.