A 1.05 liter flask contains a certain quantity of ideal gas at 305 K. Then an equal number of molecules of the same gas is added to the flask, after which the absolute pressure is 1.55 times its original value. What is the final temperature?
To solve this problem, we can use the ideal gas law, which states:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles
R = gas constant
T = temperature
Given:
Initial volume (V1) = 1.05 L
Initial temperature (T1) = 305 K
Final pressure (P2) = 1.55 * initial pressure
n1 = n2 (Equal number of molecules added)
We can rewrite the ideal gas law as:
P1V1 = n1RT1 -- Equation (1)
P2V2 = n2RT2 -- Equation (2)
Since the number of moles (n) is the same before and after, we can equate n1 = n2:
n1 = n2
Substituting the values in Equations (1) and (2) and equating n1 = n2, we get:
P1V1 = P2V2 -- Equation (3)
Now, we need to solve for T2, the final temperature. Rearranging Equation (3), we have:
V1 / V2 = P2 / P1
Simplifying:
V2 / V1 = P1 / P2
Since V2 = V1, we can write:
1 = P1 / P2
Substituting the given values:
1 = P1 / (1.55 * P1)
Cross-multiplying:
1.55 * P1 = P1
Dividing both sides by P1:
1.55 = 1
This is not possible, so there seems to be an error in the problem statement. Please recheck the values provided.
To solve this problem, we can use the ideal gas law:
PV = nRT
Where:
- P is the pressure of the gas
- V is the volume of the gas
- n is the number of moles of gas
- R is the ideal gas constant
- T is the temperature of the gas
We are given the initial temperature (T1 = 305 K) and the final pressure (P2 = 1.55P1, where P1 is the initial pressure).
Let's break down the problem step-by-step to find the final temperature (T2).
Step 1: Find the initial pressure (P1)
Since we are only given the final pressure (P2) in terms of the initial pressure (P1), we need to find the value of P1. To do this, we need to use the given information that the volume (V) of the flask is constant, and the initial number of molecules is equal to the final number of molecules.
So, the initial pressure (P1) is the pressure of the gas in the flask initially.
Step 2: Calculate the number of moles of gas in the flask initially.
Since an equal number of molecules of the same gas is added to the flask, the number of moles of gas initially is equal to the number of moles of gas finally.
Step 3: Calculate the final pressure (P2)
We are given that the final pressure (P2) is 1.55 times the initial pressure (P1).
Step 4: Use the ideal gas law equation to find T2
Plug in the values into the ideal gas law equation:
P2V = nRT2
Since the number of moles (n) and the volume (V) are the same before and after the addition of the gas:
P1V = nRT1
Therefore, we can rewrite the equation as:
P2V = P1V = nRT1
To find T2, we can rearrange the equation:
T2 = (P2V) / (nR)
Now, let's solve the problem using the given values.
Step 1: Find the initial pressure (P1)
As mentioned, P1 is unknown and needs to be determined.
Step 2: Calculate the number of moles of gas in the flask initially.
Since the number of moles of gas initially is equal to the number of moles of gas finally, this value is the same before and after the addition of the gas.
Step 3: Calculate the final pressure (P2)
Given that the final pressure (P2) is 1.55 times the initial pressure (P1).
Step 4: Use the ideal gas law equation to find T2
Plug in the values into the ideal gas law equation:
T2 = (P2V) / (nR)
By following these steps, you should be able to calculate the final temperature (T2).