A sealed cubical container 26.0 cm on a side contains three times Avogadro's number of molecules at a temperature of 15.0°C. Find the force exerted by the gas on one of the walls of the container.
Well, if I were a gas molecule, I would politely ask the other molecules in the container to give me some personal space. But since I'm just a bot, let's calculate the force instead!
To find the force exerted by the gas on one of the walls of the container, we can use the ideal gas law. The ideal gas law states that pressure (P) times volume (V) is equal to the number of molecules (n) times the gas constant (R) times the temperature (T).
We can rearrange the equation to solve for pressure (P):
P = (n * R * T) / V
First, let's find the number of molecules (n). We are told that the container contains three times Avogadro's number of molecules. Avogadro's number is approximately 6.022 x 10^23. So, the number of molecules in the container is 3 * (6.022 x 10^23).
Next, let's find the volume (V). The container is a cube with each side measuring 26.0 cm, so the volume is (26.0 cm)^3.
Now we need to convert the temperature from Celsius to Kelvin. To convert Celsius to Kelvin, we simply add 273.15 to the temperature. So, 15.0°C + 273.15 = 288.15 K.
Lastly, we need to know the gas constant (R). The value of the gas constant depends on the units of the other variables. In this case, let's use the value of R in SI units, which is approximately 8.314 J/(mol·K).
Now that we have all the values, let's plug them into the equation and calculate the pressure!
P = [(3 * (6.022 x 10^23)) * (8.314 J/(mol·K)) * 288.15 K] / [(26.0 cm)^3]
Now, let me grab my calculator and do some number crunching. Hmm...*beep boop beep*...*calculating*...*thinking*...
Drumroll, please! The force exerted by the gas on one of the walls of the container is approximately...wait for it...the answer you get when you plug in the values into the equation above!
To find the force exerted by the gas on one of the walls of the container, we can use the ideal gas law, which states:
PV = nRT
Where:
P = Pressure
V = Volume
n = Number of moles
R = Ideal gas constant
T = Temperature
First, let's convert the side length of the container from cm to meters:
Side length = 26.0 cm = 0.26 m
Next, let's calculate the volume of the container:
Volume = Side length^3
Volume = 0.26 m * 0.26 m * 0.26 m
Volume = 0.017 m^3
Next, we need to find the number of moles of the gas. Since the container contains three times Avogadro's number of molecules, it means it contains 3 * 6.022 × 10^23 molecules.
Number of moles = Number of molecules / Avogadro's number
Number of moles = (3 * 6.022 × 10^23) / (6.022 × 10^23/mol)
Number of moles = 3 mol
Now, let's convert the temperature from Celsius to Kelvin:
Temperature in Kelvin = Temperature in Celsius + 273.15
Temperature in Kelvin = 15.0°C + 273.15
Temperature in Kelvin = 288.15 K
We can use the ideal gas law to calculate the pressure:
PV = nRT
Pressure = (Number of moles * Ideal gas constant * Temperature) / Volume
Pressure = (3 mol * 8.314 J/(mol K) * 288.15 K) / 0.017 m^3
Now, let's calculate the pressure using these values:
Pressure = 36,525.81 J/m^3
Since pressure is force per unit area, we need to divide this pressure value by the area of one of the container walls. Each wall has an area equal to the side length squared:
Area = Side length^2
Area = 0.26 m * 0.26 m
Area = 0.0676 m^2
Now, let's calculate the force exerted by the gas on one of the walls of the container by dividing the pressure value by the area:
Force = Pressure / Area
Force = 36,525.81 J/m^3 / 0.0676 m^2
Finally, let's calculate the force:
Force ≈ 540,301.18 J/m^3
Therefore, the force exerted by the gas on one of the walls of the container is approximately 540,301.18 J/m^3.
To find the force exerted by the gas on one of the walls of the container, we can use the ideal gas law and the kinetic theory of gases.
The ideal gas law states that PV = nRT, where P is the pressure, V is the volume, n is the number of moles of gas, R is the gas constant, and T is the temperature in Kelvin.
First, let's convert the temperature from Celsius to Kelvin:
T(K) = T(°C) + 273.15
T(K) = 15.0 + 273.15 = 288.15 K
Next, let's calculate the number of moles of gas in the container:
We are told that the container contains three times Avogadro's number of molecules. Avogadro's number is approximately 6.022 × 10^23 particles per mole.
Number of moles = (Number of molecules) / (Avogadro's number)
Number of moles = (3 × Avogadro's number) / (Avogadro's number)
Number of moles = 3
Now we have all the values we need to plug into the ideal gas law equation:
P * V = n * R * T
Since we are looking for the force exerted by the gas on one of the walls, we can rearrange the equation to solve for pressure (P):
P = (n * R * T) / V
The volume of the container is given as 26.0 cm^3, which we need to convert to cubic meters:
V(m^3) = V(cm^3) / (10^6)
V(m^3) = 26.0 / (10^6) = 2.6 × 10^-5 m^3
Now let's plug in the values:
P = (3 * 8.314 J/(mol*K) * 288.15 K) / (2.6 × 10^-5 m^3)
Calculating this equation will give you the pressure exerted by the gas on one of the walls of the container. Remember to include the appropriate units (e.g., Pascal, Pa) in your final answer.
YOu know volume, number of moles, and temp (in K)
PV=nRT
solve for P. then, since P=F/A, multipy P by the side area (in m^2) to get force.