This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
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To calculate the final temperature of oxygen, we need to determine the polytropic index n and assume an initial temperature, as it is not provided in the problem. Step 1: Calculate the specific gas constant R for oxygen. The molar mass of oxygen (M) is 32 kg/kmol. The universal gas constant (R_u) is 8.314 kJ/(kmol·K). R = (R_u)/(M) = 8.314 kJ/(kmol·K)32 kg/kmol = 0.2598125 kJ/(kg·K) Step 2: Calculate the specific heat at constant pressure C_p and the adiabatic index k. Given C_v = 0.649 kJ/(kg·K). For a perfect gas, C_p - C_v = R. C_p = C_v + R = 0.649 kJ/(kg·K) + 0.2598125 kJ/(kg·K) = 0.9088125 kJ/(kg·K) The adiabatic index k is: k = (C_p)/(C_v) = 0.9088125 kJ/(kg·K)0.649 kJ/(kg·K) = 1.4003274 Step 3: Determine the polytropic index n. The first law of thermodynamics for a closed system is U = Q + W_on, where W_on is work done on the system and Q is heat added to the system. Work input (W_in) is done on the system, so W_on = W_in. Heat rejected (Q_rej) means heat leaves the system, so Q = -Q_rej. The problem states that "one-third of work input is rejected as heat", so Q_rej = (1)/(3) W_in. Therefore, Q = -(1)/(3) W_in. Substituting into the first law: U = -(1)/(3) W_in + W_in = (2)/(3) W_in For a perfect gas, U = m C_v (T_2 - T_1). For a polytropic process, work input W_in = (m R (T_2 - T_1))/(n-1). Equating these expressions: m C_v (T_2 - T_1) = (2)/(3) (m R (T_2 - T_1))/(n-1) Assuming T_2 ≠ T_1, we can simplify to: C_v = (2)/(3) (R)/(n-1) Rearranging to solve for n: n-1 = (2R)/(3C_v) n = 1 + (2R)/(3C_v) = 1 + 2 × 0.2598125 kJ/(kg·K)3 × 0.649 kJ/(kg·K) = 1 + (0.519625)/(1.947) = 1 + 0.2668849512 ≈ 1.2669 Step 4: Assume an initial temperature T_1. The problem does not provide an initial temperature. We will assume a standard room temperature of 25^. T_1 = 25^ = 25 + 273.15 = 298.15 K Step 5: Calculate the final temperature T_2. For a polytropic process, the relation between temperature and pressure is: (T_2)/(T_1) = ((P_2)/(P_1))^(n-1)/(n) Given P_1 = 1.05 bar and P_2 = 4.2 bar. First, calculate the exponent (n-1)/(n): (n-1)/(n) = (0.2668849512)/(1.2668849512) = 0.2106504854 Now, substitute the values into the temperature-pressure relation: (T_2)/(298.15 K) = (4.2 bar1.05 bar)^0.2106504854 (T_2)/(298.15 K) = (4)^0.2106504854 (4)^0.2106504854 ≈ 1.339106 T_2 = 298.15 K × 1.339106 T_2 ≈ 399.243 K The final temperature of oxygen is approximately 399.24 K. Drop the next question!