A cubic box of volume 4.1×10^-2 m^3 is filled with air at atmospheric pressure at 20 degrees C. The box is closed and heated to 181 degrees C. What is the net force on each side of the box?
To find the net force on each side of the box, we need to calculate the difference in pressure between the initial and final states.
Step 1: Determine the initial pressure
Given that the box is filled with air at atmospheric pressure, we can assume the initial pressure is equivalent to the atmospheric pressure. At sea level, the atmospheric pressure is typically around 101,325 Pascals (Pa).
Step 2: Convert the volume to cubic meters
The given volume is 4.1×10^-2 m^3. Since the SI unit for volume is cubic meters, no conversion is needed in this case.
Step 3: Apply the ideal gas law to find the initial number of moles
The ideal gas law equation is: PV = nRT
Where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant (8.314 J/(mol·K)), and T is the temperature in Kelvin.
Solving for n, we have: n = PV / RT
Since we don't have the number of moles (n), we can assume the air inside the box is an ideal gas and use the ideal gas law to find n. We'll need to convert the temperature from Celsius to Kelvin by adding 273.15.
Using the given temperature of 20 degrees Celsius:
T_initial = 20 + 273.15 = 293.15 K
Now we can plug in the values to find the initial number of moles:
n_initial = (P_initial * V) / (R * T_initial)
Step 4: Determine the final pressure
In this case, we need to consider the final temperature of the air inside the closed box, which is heated to 181 degrees Celsius. Again, we need to convert this temperature to Kelvin:
T_final = 181 + 273.15 = 454.15 K
Step 5: Apply the ideal gas law to find the final number of moles
Using the final temperature, we can calculate the final number of moles of air in the box:
n_final = (P_final * V) / (R * T_final)
Step 6: Calculate the pressure difference
The net force on each side of the box can be determined by finding the difference between the final and initial pressures:
ΔP = P_final - P_initial
Step 7: Substitute the values into the equation
Substitute the values of n_initial, n_final, and V into the equation:
ΔP = (n_final * R * T_final / V) - (n_initial * R * T_initial / V)
Step 8: Calculate the net force
Since the net force is the same on each side of the box, we can divide the pressure difference by the area of one side of the box.
Net force = ΔP / A
Note: The area of one side of a cubic box is given by A = L^2, where L is the length of one side of the box.
By following these steps, you should be able to calculate the net force on each side of the box.
The edge size of the box is
a = (0.041)^(1/3) = 0.345 m
and the area of a side is 0.1189 m^2
Pressure increases by a factor (273+181)/(273+20) = 1.5495
The new pressure will be 1.5495 atm
Multiply the final pressure (in Pascals) by the box side area, for the force in Newtons. I will leave that step up to you.