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
Alright Ðƴŋâmïč̣ — let's do this. Step 1: Identify the given values and convert temperatures to the absolute Kelvin scale. Initial conditions: • Initial gauge pressure, P_1,gauge = 2.202 × 10^5 Pa. • Initial volume, V_1 = 4 m^3. • Initial temperature, T_1 = 27^ C. Convert to Kelvin: T_1 = 27 + 273 = 300 K. Final conditions: • Final volume, V_2 = 0.025 m^3. • Final temperature, T_2 = 40^ C. Convert to Kelvin: T_2 = 40 + 273 = 313 K. We need to calculate the final gauge pressure, P_2,gauge. Assume standard atmospheric pressure, P_atm = 1.013 × 10^5 Pa. Step 2: Calculate the initial absolute pressure. Absolute pressure is the sum of gauge pressure and atmospheric pressure. P_1,abs = P_1,gauge + P_atm P_1,abs = 2.202 × 10^5 Pa + 1.013 × 10^5 Pa P_1,abs = (2.202 + 1.013) × 10^5 Pa P_1,abs = 3.215 × 10^5 Pa Step 3: Apply the Combined Gas Law. The Combined Gas Law relates pressure, volume, and temperature for a fixed amount of gas: (P_1 V_1)/(T_1) = (P_2 V_2)/(T_2) We need to solve for the final absolute pressure, P_2,abs. Rearrange the formula: P_2,abs = P_1,abs V_1 T_2T_1 V_2 Step 4: Substitute the values and calculate P_2,abs. P_2,abs = (3.215 × 10^5 Pa) × (4 m^3) × (313 K)(300 K) × (0.025 m^3) P_2,abs = 4024.58 × 10^5 Pa7.5 P_2,abs = 536.61066... × 10^5 Pa Step 5: Convert the final absolute pressure to gauge pressure. P_2,gauge = P_2,abs - P_atm P_2,gauge = (536.61066 × 10^5 Pa) - (1.013 × 10^5 Pa) P_2,gauge = (536.61066 - 1.013) × 10^5 Pa P_2,gauge = 535.59766 × 10^5 Pa Rounding to three significant figures, consistent with the input values: P_2,gauge ≈ 5.36 × 10^7 Pa The final gauge pressure is 5.36 × 10^7 Pa. What's next?