A balloon filled with 22.3 mol helium has a volume of 500. L at 0.0°C and 1.00 atm pressure. The temperature of the balloon is increased to 38.0°C as it expands to a volume of 570. L, the pressure remaining constant. Calculate q, w, and E for the helium in the balloon. (The molar heat capacity for helium gas is 20.8 J/°C·mol.)
Answered a later post.
To calculate q (heat), w (work), and E (internal energy), we can use the First Law of Thermodynamics, also known as the energy balance equation:
ΔE = q - w
where ΔE is the change in internal energy, q is the heat transferred to the system, and w is the work done by the system.
Let's break down the problem step by step:
Step 1: Calculate the change in internal energy (ΔE)
ΔE = E_final - E_initial
Step 2: Calculate the heat transferred (q)
q = n * C * ΔT
where n is the number of moles, C is the molar heat capacity, and ΔT is the change in temperature.
Step 3: Calculate the work done (w)
w = -P * ΔV
where P is the pressure and ΔV is the change in volume.
Now let's compute the solution:
Step 1: Calculate the change in internal energy (ΔE)
ΔE = E_final - E_initial
Since the problem asks for the change in energy, we do not need to calculate the specific values for E_final and E_initial.
Step 2: Calculate the heat transferred (q)
q = n * C * ΔT
Given:
n = 22.3 mol (number of moles)
C = 20.8 J/°C·mol (molar heat capacity)
ΔT = (38.0°C - 0.0°C) = 38.0°C (change in temperature)
q = 22.3 mol * 20.8 J/°C·mol * 38.0°C
= 16889.12 J
Step 3: Calculate the work done (w)
w = -P * ΔV
Given:
P = 1.00 atm (pressure)
ΔV = (570. L - 500. L) = 70. L (change in volume)
Note that the pressure remains constant, so the work done is simply the product of the pressure and the change in volume.
Since the pressure is given in atm, we need to convert the volume from liters (L) to cubic meters (m^3) to use the standard unit of pressure (Pa):
1 L = 0.001 m^3
ΔV = 70. L * 0.001 m^3/L
= 0.07 m^3
Now we can calculate the work done:
w = -P * ΔV
= -(1.00 atm * 0.07 m^3)
= - (1.00 atm * 0.07 m^3 * 101.325 J/atm)
= -7.089 J
Step 4: Calculate the change in internal energy (ΔE)
ΔE = q - w
= 16889.12 J - (-7.089 J)
= 16896.209 J
Thus, the values for q, w, and E for the helium in the balloon are as follows:
q = 16889.12 J
w = -7.089 J
E = 16896.209 J