Some incandescent light bulbs are filled with argon gas. What is vrms for argon atoms near the filament, assuming their temperature is 2100 K?
To determine the root mean square velocity (vrms) of argon atoms near the filament in an incandescent light bulb, we need to make use of the kinetic theory of gases and apply the ideal gas law.
The ideal gas law is expressed as:
PV = nRT
Where:
P is the pressure of the gas
V is the volume it occupies
n is the number of moles of gas
R is the ideal gas constant
T is the temperature of the gas in Kelvin
In this case, we need to find the vrms, which is related to the average kinetic energy of the gas particles. The average kinetic energy (KE) of a gas particle can be calculated using the following formula:
KE = (3/2) * k * T
Where:
k is the Boltzmann constant (1.38 * 10^-23 J/K)
T is the temperature in Kelvin
The kinetic energy of a gas particle is directly related to its velocity. The relationship is given by:
KE = (1/2) * m * v^2
Where:
m is the mass of a single gas particle
v is the velocity of the gas particle
Rearranging the equation, we can solve for v:
v = √((2 * KE) / m)
Now let's plug in the values to find vrms for argon near the filament, assuming a temperature of 2100 K.
First, we need to determine the average kinetic energy:
KE = (3/2) * k * T
= (3/2) * (1.38 * 10^-23 J/K) * (2100 K)
≈ 4.185 * 10^-20 J
Next, we need to find the mass of a single argon atom. The molar mass of argon (Ar) is approximately 39.95 g/mol. To convert this to kilograms (kg) and find the mass of a single argon atom, we use Avogadro's number (6.022 * 10^23 atoms/mol).
Mass of a single Ar atom = (39.95 g/mol) / (6.022 * 10^23 atoms/mol)
= 6.645 * 10^-23 g
Converting to kilograms:
Mass of a single Ar atom = (6.645 * 10^-23 g) / (1000 g/kg)
≈ 6.645 * 10^-26 kg
Now, we can calculate the vrms:
v = √((2 * KE) / m)
= √((2 * 4.185 * 10^-20 J) / (6.645 * 10^-26 kg))
≈ 1574 m/s
Therefore, the vrms for argon atoms near the filament in an incandescent light bulb at a temperature of 2100 K is approximately 1574 m/s.