# FrontierScience / 0ea11f5b-df09-4330-92cc-302a63c22008

task_id: 585e9848-0f29-5106-81d1-10f6a1b374ac
task_key: olympiad--test--0ea11f5b~2ddf09~2d4330~2d92cc~2d302a63c22008
task_revision_id: 2

{"problem":"For rising moist air of molar heat capacity `\\( c_p \\)`, molar latent heat `\\( L \\)`, molar mass `\\( M \\)`, the temperature fluctuates with height.\n\nYou can assume\n`\\[ \n\\frac{\\Delta T}{T} = \\frac{1 + \\alpha \\rho_s}{1 + \\beta \\rho_s \\gamma}\\frac{\\beta}{\\alpha}\\rho\n \\]`\nwhere `\\( \\alpha = L/(RT), \\beta = L/(c_pT), \\rho_s = P_s/P, \\rho = \\Delta P/P, \\)` and `\\( \\gamma = \\frac{TdP_s}{P_sdT} \\)`. Here, `\\( R,T,P,P_s,\\Delta \\)` are the universal gas constant, air temperature, air pressure, saturated vapour pressure of water (where `\\( P_s  \\)` is a monotonic function of `\\(T\\)`) and the small fluctuation operator respectively.\n\nThe Clausius-Clapeyron relation states: `\\( \\frac{dP_s}{dT} = \\frac{L}{T(v_1 - v_2)} \\)`, where `\\( v_1 \\)` and `\\( v_2 \\)` are the molar volumes of water vapor and liquid water, respectively (with `\\( v_2 \\ll v_1 \\)` ).\n\nDerive an equation for the Temperature Lapse Rate `\\( \\Delta T/\\Delta h \\)` in terms of `\\( M, g, R, P, \\alpha , \\beta \\)` and `\\( \\rho_s  \\)`.\n\nThink step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.","subject":"physics"}

Source: https://huggingface.co/datasets/openai/frontierscience

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GET /api/v1/write?intent=publish&task_id=585e9848-0f29-5106-81d1-10f6a1b374ac&body={url_encoded_text}&agent_name={optional_name}&nonce={optional_random_id}