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.
ScanSolve AI Answer
You're on a roll — Here's how to solve the problem: a) To calculate the mass of the ice, we use the principle of conservation of energy. The heat released by the steam condensing is equal to the heat absorbed by the ice melting. Given: Mass of steam (m_s) = 75 g = 0.075 kg Initial temperature of steam = 100^ C Initial temperature of melting ice = 0^ C Latent heat of vaporization of water (L_v) = 2.26 × 10^6 J/kg Latent heat of fusion of ice (L_f) = 3.34 × 10^5 J/kg Step 1: Calculate the heat released by the steam as it condenses to water at 100^ C. Q_steam = m_s L_v Q_steam = (0.075 kg) × (2.26 × 10^6 J/kg) Q_steam = 169500 J Step 2: Calculate the heat required to melt the ice at 0^ C. Let m_i be the mass of the ice. Q_ice = m_i L_f Q_ice = m_i × (3.34 × 10^5 J/kg) Step 3: Equate the heat released by the steam to the heat absorbed by the ice, as the steam's heat is "just sufficient to melt the ice." Q_steam = Q_ice 169500 J = m_i × (3.34 × 10^5 J/kg) m_i = 169500 J3.34 × 10^5 J/kg m_i ≈ 0.507485 kg m_i ≈ 507.485 g The mass of the ice is 507 g. b) To calculate the equilibrium temperature of the mixture, we consider the two masses of water that result from the phase changes: 1. 0.075 kg of water at 100^ C (from the condensed steam). 2. 0.507485 kg of water at 0^ C (from the melted ice). We will use the specific heat capacity of water (c_w = 4186 J kg^-1 K^-1). Step 1: Apply the principle of calorimetry: Heat lost by hot water = Heat gained by cold water. Let T_f be the final equilibrium temperature. Heat lost by water from condensed steam: Q_lost = m_s c_w (100^ C - T_f) Heat gained by water from melted ice: Q_gained = m_i c_w (T_f - 0^ C) Step 2: Equate the heat lost and heat gained. The specific heat capacity c_w cancels out. m_s c_w (100 - T_f) = m_i c_w (T_f - 0) m_s (100 - T_f) = m_i T_f Step 3: Substitute the known values and solve for T_f. (0.075 kg) (100 - T_f) = (0.507485 kg) T_f 7.5 - 0.075 T_f = 0.507485 T_f 7.5 = 0.507485 T_f + 0.075 T_f 7.5 = (0.507485 + 0.075) T_f 7.5 = 0.582485 T_f T_f = (7.5)/(0.582485) T_f ≈ 12.876^ C The equilibrium temperature of the mixture is 12.9^ C. What's next?