2. A Sample of V1=6.2L of pure nitrogen gas at 21.85 degrees Celsius and total pressure (gauge plus ambient) of p1=155,700 Pa is heated so that the vrms for the gas molecules increase by 9%. Find the new absolute temperature for the gas, and compute the pressure for the bottled gas in the new thermodynamic state.
To answer this question, we need to use the ideal gas law and the equation for the root mean square velocity (vrms) of a gas molecule.
The ideal gas law equation is:
PV = nRT
Where:
P is the pressure of the gas
V is the volume of the gas
n is the number of moles of the gas
R is the ideal gas constant
T is the absolute temperature of the gas
The equation for the root mean square velocity (vrms) is:
vrms = sqrt((3RT) / M)
Where:
R is the ideal gas constant
T is the absolute temperature of the gas
M is the molar mass of the gas
1. Calculate the initial number of moles (n1) using the ideal gas law:
P1V1 = n1RT1
Since we are given the volume (V1), pressure (P1), and temperature (T1) of the initial state, we can rearrange the equation to solve for n1:
n1 = (P1V1) / (RT1)
2. Calculate the initial molar mass of nitrogen gas (M) using its atomic mass:
M = (2 * atomic mass of nitrogen) = (2 * 14 g/mol) = 28 g/mol
3. Calculate the initial absolute temperature (T1) using the given temperature in Celsius:
T1 = 21.85 + 273.15 = 295 K
4. Calculate the initial root mean square velocity (vrms1) using the formula:
vrms1 = sqrt((3RT1) / M)
5. Calculate the final root mean square velocity (vrms2) using the equation:
vrms2 = vrms1 * (1 + 0.09) = vrms1 * 1.09
6. Calculate the final absolute temperature (T2) using the formula for vrms and the given vrms2:
vrms2 = sqrt((3RT2) / M)
Rearranging the equation, we get:
T2 = (vrms2^2 * M) / (3R)
7. Finally, calculate the final pressure (P2) using the ideal gas law equation:
P2 = (n1RT2) / V1
Now you can plug in the values and calculate the new absolute temperature (T2) and the pressure (P2) for the gas in its new thermodynamic state.