a) Which is the greater speed, that of a bullet fired from a high powered M-16 rifle (2180 mi/hr) or the root mean square speed of H2 molecules at 25 degrees Celsius.
I completed part a) however I don't know how to answer part b)
b) At what temperature are the Urms of H2 and the speed of the M-16 rifle bullet given in part a) the same?
KE=3/2 kb*Temp
so if you solved the first part, you must have used the formula to find the KE of the molecules (1/2 m v^2).
So use the same formula to find Temp when v=speed of bullet. (change to m/s)
To solve part b), we need to equate the root mean square speed of H2 molecules and the speed of the M-16 rifle bullet from part a).
Given:
Speed of M-16 bullet = 2180 mi/hr
We need to find the temperature at which the root mean square speed (Urms) of H2 molecules is equal to the speed of the M-16 bullet.
The root mean square speed (Urms) of a gas particle is given by the equation:
Urms = √(3 * k * T / m)
Where:
k is the Boltzmann constant (1.38 * 10^-23 J/K)
T is the temperature in Kelvin
m is the molar mass of H2 (2 g/mol)
To find the temperature at which the Urms of H2 and the speed of the M-16 rifle bullet are the same, we need to equate the equations:
Urms = Speed of M-16 bullet
√(3 * k * T / m) = 2180 mi/hr
First, we need to convert the speed of the bullet from miles per hour to meters per second for consistency:
1 mile = 1609.34 meters
1 hour = 3600 seconds
Speed of M-16 bullet = 2180 mi/hr * (1609.34 meters/1 mile) * (1 hour/3600 seconds)
Speed of M-16 bullet = 975.7 meters/second
Now, we can square both sides of the equation:
3 * k * T / m = (975.7)^2
Simplify:
3 * k * T = [(975.7)^2 * m]
Solve for T:
T = [(975.7)^2 * m] / (3 * k)
Substitute the values:
T = [(975.7)^2 * 2] / (3 * 1.38 * 10^-23)
T ≈ 5.9167 * 10^7 K
Therefore, at a temperature of approximately 5.9167 * 10^7 Kelvin, the root mean square speed of H2 molecules and the speed of the M-16 rifle bullet from part a) are the same.
To answer part b), we need to compare the root mean square speed (Urms) of H2 molecules to the speed of the bullet fired from the M-16 rifle.
First, let's calculate the Urms of H2 molecules at 25 degrees Celsius. To do this, we can use the root mean square speed formula:
Urms = √(3RT/M)
Where:
- R is the ideal gas constant (8.314 J/(mol·K))
- T is the temperature in Kelvin
- M is the molar mass of H2 (2 g/mol)
Converting 25 degrees Celsius to Kelvin:
T = 25 + 273.15 = 298.15 K
Calculating the Urms for H2:
Urms = √(3 * 8.314 J/(mol·K) * 298.15 K / 2 g/mol)
= √(7463.887 J/mol)
≈ 86.39 m/s
Now, let's compare this to the speed of the bullet fired from the M-16 rifle, which is given as 2180 mi/hr. We need to convert this speed to meters per second (m/s):
1 mile = 1609.34 meters
1 hour = 3600 seconds
Speed of bullet = 2180 mi/hr * (1609.34 meters / 1 mile) * (1 hour / 3600 seconds)
≈ 976.85 m/s
Now, we want to find the temperature at which the Urms of H2 and the speed of the bullet are equal. We'll assume that the ideal gas equation applies for H2 gas:
Urms = √(3RT/M)
Setting the two speeds equal to each other:
86.39 m/s = 976.85 m/s
Since the speed of the bullet is much higher than the Urms of H2 molecules, it is safe to assume that the speed of the bullet will never be equal to the Urms of H2 molecules. Therefore, there is no temperature at which the two speeds are the same.
In conclusion, there is no temperature at which the Urms of H2 molecules and the speed of the bullet fired from the M-16 rifle are equal.