An experiment makes use of a water manometer attached to a flask. Initially the two columns in the gas manometer are at the same level and the air pressure in the flask and both sides of manometr is 1 atm. The experiment is set up when the air pressure is 7 degree Celcius. The left side of the manometer is connected to a flask and right side is capped so that the air at the end will be compressed when the flask is heated by a gas burner. The cap is initially 15cm above the water column. The volume of the flask is 1*10^4 m^3. When calculating the change in pressure assosiated with the heating of the gas in the flask, you can neglect the change in the volume of the gas(air in this cae) assosiated with the displacement of the water column in the manometer. Calculate how many calories (cal) have been added to the flask through heating from the gas burner given that specific heat of the air is 20.8 (J/K)/mol.
To calculate the calories (cal) added to the flask through heating, we need to determine the change in pressure associated with the heating of the gas in the flask and then use this information to calculate the heat transferred.
Step 1: Determine the change in pressure
The change in pressure can be determined by calculating the difference in the water levels in the manometer before and after heating.
Given:
- Initial air pressure in the flask and both sides of manometer = 1 atm
- Temperature at the start of the experiment = 7°C
By using the ideal gas law, we can convert the initial air pressure to absolute pressure:
P1 = 1 atm + atmospheric pressure at 7°C (which can be looked up in a table)
Step 2: Calculate the change in pressure
Let's assume the final air pressure is P2 (after heating).
Using the equation for the change in pressure due to a column of fluid in a manometer:
ΔP = ρgh
Where:
ΔP = change in pressure
ρ = density of the fluid (water)
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height difference between the two columns
In this case, the height difference h is equal to the initial height of the cap above the water column (15 cm).
Step 3: Convert the change in pressure to calories
To convert the change in pressure to calories, we can use the following conversion:
1 J = 0.239006 calories
We already know the specific heat of air, so we can calculate the change in internal energy (ΔU) using the equation:
ΔU = nCΔT
Where:
ΔU = change in internal energy
n = number of moles of air
C = specific heat of air
ΔT = change in temperature
Finally, we can convert the change in internal energy from Joules to calories:
1 cal = 4.18 J
Note: It's important to convert all units to the same system (SI) before performing calculations.
I will provide the step-by-step calculation in the following response.
To calculate the calories added to the flask through heating, we need to determine the change in pressure caused by the increase in temperature.
First, let's convert the Celsius temperature to Kelvin. We can use the equation:
K = °C + 273.15
So, the temperature in Kelvin would be:
T = 7°C + 273.15 = 280.15 K
Since the volume of the gas in the flask remains constant, we can use the ideal gas law to relate the change in pressure to the change in temperature:
ΔP = (n * R * ΔT) / V
Where:
ΔP is the change in pressure,
n is the number of moles of gas,
R is the ideal gas constant (8.314 J/(mol·K)),
ΔT is the change in temperature,
and V is the volume of the flask.
First, we need to determine the number of moles of gas (air in this case) in the flask. We can use the ideal gas equation:
PV = nRT
Since the pressure and volume are given at the initial state, we can rearrange the equation to solve for n:
n = (PV) / (RT)
Substituting the given values, we get:
n = (1 atm * 1 * 10^4 m^3) / (8.314 J/(mol·K) * 280.15 K)
Now, we can substitute the value of n into the first equation to calculate the change in pressure:
ΔP = (n * R * ΔT) / V
ΔP = [(1 atm * 1 * 10^4 m^3) / (8.314 J/(mol·K) * 280.15 K)] * (8.314 J/(mol·K)) * (280.15 K - 273.15 K) / (1 * 10^4 m^3)
Simplifying the equation gives us the change in pressure, ΔP.
Finally, we can calculate the calories added to the flask through heating using the formula:
Calories = ΔP * V * specific heat
Calories = ΔP * (1 * 10^4 m^3) * (20.8 (J/K)/mol)
Simplifying the equation will give us the number of calories added to the flask through heating.