You have a sample of gas at -30 C. You wish to increase the rms speed by 12%. To what temperature should the gas be heated?
rms means the square root of the mean squared.
So increasing rms by 1.12 means increasing energy by 1.12 squared, or 1.25
But temp is a measure of avg KE, or avg v^2, so temp should increase by 1.25
The original temp was 273-30K, or 243, so the new tempwill be 305K.
To find the temperature to which the gas should be heated, we first need to understand the relationship between temperature and the root mean square (rms) speed of gas particles.
The root mean square speed of gas particles can be determined using the following equation:
v = √(3kT / m)
where:
v is the root mean square speed
k is the Boltzmann constant (1.38 × 10^-23 J/K)
T is the temperature in Kelvin
m is the molar mass of the gas in kg
In this case, we are given the initial temperature of the gas (-30°C), which needs to be converted into Kelvin before substituting it into the equation. The formula to convert Celsius to Kelvin is:
T(K) = T(°C) + 273.15
So, converting the initial temperature:
T_initial = -30 + 273.15 = 243.15 K
Now, since we want to increase the rms speed by 12 percent, we can multiply the initial rms speed by 1.12 to get the new rms speed:
v_new = 1.12 * v_initial
To find the new temperature, we rearrange the equation to solve for T:
T_new = (v_new^2 * m) / (3k)
Now we have all the information we need to calculate the new temperature. Substituting the values into the equation:
T_new = (v_initial^2 * m * 1.12^2) / (3k)
The molar mass of the gas is needed to proceed with the calculation. If you provide the molar mass, I can calculate the new temperature for you.
To determine the temperature to which the gas should be heated, we can use the formula for the root mean square (rms) speed of gas molecules:
v = sqrt(3kT/m)
Where:
- v is the rms speed of gas molecules,
- k is the Boltzmann constant (1.38 x 10^-23 J/K),
- T is the temperature in Kelvin, and
- m is the molar mass of the gas.
First, we need to convert the given temperature of -30 °C to Kelvin:
T_initial = -30 °C + 273.15 = 243.15 K
Next, we need to calculate the new rms speed, v_new, by increasing the initial rms speed by 12%:
v_new = v_initial + 0.12 * v_initial
v_new = 1.12 * v_initial
To find the new temperature T_new, we'll assume the molar mass and k remain constant:
1.12 * v_initial = sqrt(3kT_new/m)
Rearranging the equation:
(1.12 * v_initial)^2 = 3kT_new/m
Substituting the values:
(1.12 * sqrt(3kT_initial/m))^2 = 3kT_new/m
1.2544 * (3kT_initial/m) = 3kT_new/m
Simplifying the equation:
1.2544 * T_initial = T_new
Finally, we substitute T_initial = 243.15 K into the equation to find T_new:
T_new = 1.2544 * 243.15 K = 305.6 K
Therefore, the gas should be heated to approximately 305.6 K to increase the root mean square speed by 12%.