An ideal gas at 18.0 °C and a pressure of 2.83 x 105 Pa occupies a volume of 2.63 m3. (a) How many moles of gas are present? (b) If the volume is raised to 5.09 m3 and the temperature raised to 31.6 °C, what will be the pressure of the gas?
If N is the number of molecules then:
PV = NkT ---->
N = PV/(k T)
P = of 2.83 x 10^5 Pa
T = 291.15 K
V = 2.63 m^3
k = 1.38065*10^(-23)J K^(-1)
Calculate N and divide by 1 mole.
1 mole times the mass of one atomic mass unit is 1 gram.
1 atomic mass unit is 1.66054*10^(-27) kg ---->
1 mole is 6.02214*10^(23)
Remember: a mole is just a number just like a billion is a number!
To solve part (a) of the question and find the number of moles of gas present, we can use the ideal gas law equation PV = NkT, where P is the pressure, V is the volume, N is the number of molecules, k is the Boltzmann constant, and T is the temperature in Kelvin.
Given:
P = 2.83 x 10^5 Pa
V = 2.63 m^3
T = 18.0 °C = 18.0 + 273.15 = 291.15 K
Substituting the given values into the equation, we have:
N = PV/(kT)
Now we need to calculate the value of k, which is the Boltzmann constant. The Boltzmann constant is approximately equal to 1.38065 x 10^(-23) J K^(-1).
Next, we will calculate N by substituting the given values into the equation:
N = (2.83 x 10^5 Pa) x (2.63 m^3)/[(1.38065 x 10^(-23) J K^(-1)) x (291.15 K)]
After evaluating the expression, we will have the number of molecules present.
To convert the number of molecules to moles, we use Avogadro's number, which is approximately equal to 6.02214 x 10^23 mol^(-1). So, by dividing the number of molecules by Avogadro's number, we can find the number of moles of gas.
Finally, since 1 mole of any substance contains the same number of entities (atoms, molecules, etc.), which is Avogadro's number, we know that the number of moles of any substance is equal to its mass in grams.
We can then conclude that the number of moles of gas present is equal to its mass in grams, which is equal to the calculated value of N divided by Avogadro's number.
Now, let's move on to solve part (b) of the question.
To find the pressure of the gas when the volume is raised to 5.09 m^3 and the temperature is raised to 31.6 °C, we can use the ideal gas law equation again:
PV = NkT
Given:
V = 5.09 m^3
T = 31.6 °C = 31.6 + 273.15 = 304.75 K
We can rearrange the equation to solve for P:
P = NkT/V
To find the new pressure, we need the value of N, the number of moles of gas. We can use the same formula we derived in part (a) to calculate N. Once we have the value of N, we substitute it, along with the known values of k, T, and V, into the equation P = NkT/V to find the pressure of the gas.
I hope this explanation helps you understand how to solve the given problem.