In a fixed volume container, 6*10^23 molecules of a gas exert a pressure of 12 N/m^2. If 12*10^23 molecules of gas were added to the container, what would the pressure become?
PV=nRT
P=pressure in atm
V=volume in L
T=absolute (kelvin) temperature
R=0.082 L atm/mole K
n=number of moles
***the answer is 36 N/m^2. I don't know how to get this answer. Thank you for the help!
36
To find the new pressure after adding 12*10^23 molecules of gas to the container, we can use the ideal gas law equation, PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the absolute temperature.
First, let's determine the number of moles of gas initially in the container. Given that there are 6*10^23 molecules, we can divide this number by Avogadro's number (6.022*10^23 molecules/mole) to get the number of moles:
n1 = (6*10^23 molecules) / (6.022*10^23 molecules/mole)
n1 = 0.997 moles (approximately)
Next, let's calculate the initial pressure in atm. Given that the pressure is 12 N/m^2, we need to convert it to atm. Since 1 atm = 101325 N/m^2:
P1 = (12 N/m^2) / (101325 N/m^2/atm)
P1 = 1.185*10^-4 atm (approximately)
Now, let's find the final number of moles after adding the additional gas. We simply add the given 12*10^23 molecules to the initial moles:
n2 = n1 + (12*10^23 molecules) / (6.022*10^23 molecules/mole)
n2 = 0.997 + 1.993 moles (approximately)
n2 = 2.99 moles (approximately)
Finally, we can use the ideal gas law to obtain the final pressure by rearranging the equation:
P2 = (n2 * R * T) / V
Since the volume is fixed, we can assume V1 = V2. Therefore, we can write:
P1 = (n1 * R * T) / V1
P2 = (n2 * R * T) / V2
Since V1 = V2, we can simplify the equation to:
P1 = (n1 * R * T) / V
P2 = (n2 * R * T) / V
We can now calculate the final pressure:
P2 = (2.99 moles * 0.082 L atm/mole K * T) / V
Substituting P1 = 1.185*10^-4 atm and V1 = V into the equation, we can solve for P2:
P2 = (2.99 * 0.082 * T) / V
Since the ratio between P2 and P1 is proportional to the ratio between n2 and n1, and V1 = V2, we can write:
P2 / P1 = n2 / n1
Since P1 = 1.185*10^-4 atm, the equation becomes:
P2 / 1.185*10^-4 atm = 2.99 moles / 0.997 moles
Finally, rearranging the equation to solve for P2:
P2 = (2.99 moles * 1.185*10^-4 atm) / 0.997 moles
Calculating this expression:
P2 = 3.562*10^-4 atm
To convert this value to N/m^2, we multiply it by 101325 N/m^2/atm:
P2 = (3.562*10^-4 atm) * (101325 N/m^2/atm)
P2 = 36 N/m^2 (approximately)
Therefore, after adding 12*10^23 molecules of gas to the container, the pressure would become approximately 36 N/m^2.