An ideal monatomic gas is contained in a vessel of constant volume 0.230 m3. The initial temperature and pressure of the gas are 300 K and 5.00 atm, respectively. The goal of this problem is to find the temperature and pressure of the gas after 23.0 kJ of thermal energy is supplied to the gas. (a) Use the ideal gas law and initial conditions to calculate the number of moles of gas in the vessel. mol (b) Find the specific heat of the gas. J/K (c) What is the work done by the gas during this process? kJ (d) Use the first law of thermodynamics to find the change in internal energy of the gas. kJ (e) Find the change in temperature of the gas. K (f) Calculate the final temperature of the gas. K (g) Use the ideal gas expression to find the final pressure of the gas. atm
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To solve this problem, we need to apply the ideal gas law, the concept of specific heat, and the first law of thermodynamics. Let's begin with part (a) and follow through to part (g).
(a) Use the ideal gas law, PV = nRT, to calculate the number of moles (n) of the gas.
Given: - Initial pressure, P = 5.00 atm. To use SI units, we convert this to Pascals: 5.00 atm × 101325 Pa/atm = 506625 Pa - Volume, V = 0.230 m³ (constant) - Initial temperature, T = 300 K - Ideal gas constant, R = 8.314 J/(mol⋅K)
Solving for n: n = PV / RT n = (506625 Pa × 0.230 m³) / (8.314 J/(mol⋅K) × 300 K) n = 23.30 mol
(b) The specific heat at constant volume for a monatomic ideal gas is C_v = (3/2)R, where R is the ideal gas constant. C_v = (3/2) × 8.314 J/(mol⋅K) C_v = 12.471 J/(mol⋅K)
(c) Because the volume is constant, the work done by the gas is zero. Work, W, is defined as the product of pressure and the change in volume. Since volume does not change, ∆V = 0, so W = 0 kJ.
(d) Apply the first law of thermodynamics: ∆U = Q - W, where ∆U is the change in internal energy, Q is the heat supplied, and W is the work done by the system. Since W = 0 kJ, the entire 23.0 kJ of thermal energy goes into changing the internal energy of the gas. ∆U = 23.0 kJ
(e) Use the relationship between change in internal energy and specific heat at constant volume: ∆U = nC_v∆T, where ∆T is the change in temperature. ∆U = nC_v∆T 23.0 kJ = (23.30 mol)(12.471 J/(mol⋅K))∆T 23,000 J = (290.7767 J/K)∆T ∆T = 23,000 J / 290.7767 J/K ∆T = 79.1 K (rounded to three significant figures)
(f) Calculate the final temperature, T_final: T_final = T_initial + ∆T T_final = 300 K + 79.1 K T_final = 379.1 K
(g) Now use the ideal gas expression PV = nRT to find the final pressure, P_final: P_final = nRT_final / V P_final = (23.30 mol)(8.314 J/(mol⋅K))(379.1 K) / 0.230 m³ P_final = 723,285.946 Pa To convert this back to atmospheres: P_final = 723,285.946 Pa / 101325 Pa/atm P_final = 7.14 atm (rounded to three significant figures)
Summary: (a) Number of moles of gas = 23.30 mol (b) Specific heat of the gas = 12.471 J/(mol⋅K) (c) Work done by the gas = 0 kJ (d) Change in internal energy of the gas = 23.0 kJ (e) Change in temperature of the gas = 79.1 K (f) Final temperature of the gas = 379.1 K (g) Final pressure of the gas = 7.14 atm
Extra: The ideal gas law states that for a given amount of an ideal gas, the pressure, volume, and temperature are related through the equation PV = nRT. The ideal gas constant, R, depends on the units being used and is a proportionality constant that relates the energy scale to the temperature scale.
Specific heat is the amount of heat per unit mass required to raise the temperature by one degree Celsius or one Kelvin. In the context of ideal gases, specific heat at constant volume (C_v) is used when the volume of the gas does not change and specific heat at constant pressure (C_p) is used when the pressure of the gas does not change.
The first law of thermodynamics states that the change in internal energy (∆U) of a closed system is equal to the heat added to the system (Q) minus the work done by the system (W). In processes occurring at constant volume (like this one), there is no work done by the system because there is no volume change; therefore, all the heat added goes into changing the internal energy. This increase in internal energy is manifested as an increase in temperature.
The concept of internal energy is related to the kinetic energy of the gas particles. For a monatomic ideal gas, internal energy is directly proportional to temperature, and the specific heat at constant volume (C_v) is used to calculate the change in internal energy with respect to a change in temperature.