A balloon holding 4.50 moles of helium (He) gas absorbs 975 J of thermal energy while doing 127 J of work expanding to a larger volume.HINT(a) Apply the first law of thermodynamics. (b) Recall that the molar specific heat at constant volume, Cv, takes different values depending on whether the gas is monatomic or diatomic.Click the hint button again to remove this hint.(a)Find the change in the balloon's internal energy (in J). J(b)Calculate the change in temperature of the gas (in K). K
Question
A balloon holding 4.50 moles of helium (He) gas absorbs 975 J of thermal energy while doing 127 J of work expanding to a larger volume.HINT(a) Apply the first law of thermodynamics. (b) Recall that the molar specific heat at constant volume, Cv, takes different values depending on whether the gas is monatomic or diatomic.Click the hint button again to remove this hint.(a)Find the change in the balloon's internal energy (in J). J(b)Calculate the change in temperature of the gas (in K). K
Solution
(a) The first law of thermodynamics states that the change in internal energy (∆U) of a system is equal to the heat added to the system (Q) minus the work done by the system (W). Mathematically, this is represented as:
∆U = Q - W
Given that the balloon absorbs 975 J of thermal energy (Q = 975 J) and does 127 J of work (W = 127 J), we can substitute these values into the equation to find the change in internal energy:
∆U = 975 J - 127 J = 848 J
So, the change in the balloon's internal energy is 848 J.
(b) The change in temperature (∆T) of a gas can be calculated using the formula:
∆T = ∆U / (n * Cv)
where n is the number of moles of the gas and Cv is the molar specific heat at constant volume. For a monatomic gas like helium, Cv = 3/2 R, where R is the ideal gas constant (8.314 J/(mol*K)).
Substituting the given and known values into the equation:
∆T = 848 J / (4.50 mol * 3/2 * 8.314 J/(mol*K)) = 848 J / (31.617 J/K) = 26.8 K
So, the change in temperature of the gas is 26.8 K.
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