If a heater supplies 1.8 x 10⁶ J/h to a room 3.5 m x 4.6 m x 3.0 m containing air at 20°C and 1.0 atm, by how much will the temperature rise in one hour, assuming no losses of heat or air mass to the outside? Assume air is an ideal diatomic gas with molecular mass 29.

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Determine the volume of the room using the formula for the volume of a rectangular prism: \( V = \text{length} \times \text{width} \times \text{height} \). Substitute the given dimensions \( 3.5 \ \text{m} \), \( 4.6 \ \text{m} \), and \( 3.0 \ \text{m} \) to calculate \( V \).

Calculate the mass of the air in the room. Use the ideal gas law \( PV = nRT \) to find the number of moles \( n \), where \( P \) is the pressure (1.0 atm), \( V \) is the volume (from step 1), \( R \) is the ideal gas constant (8.314 J/mol·K), and \( T \) is the temperature in kelvins (convert 20°C to K by adding 273.15). Then, multiply \( n \) by the molecular mass of air (29 g/mol or 0.029 kg/mol) to find the mass.

Determine the specific heat capacity of air. Since air is an ideal diatomic gas, its molar specific heat at constant volume \( C_v \) is approximately \( 5R/2 \), where \( R \) is the ideal gas constant. Convert this to specific heat capacity per unit mass by dividing \( C_v \) by the molar mass of air.

Relate the heat supplied to the temperature change using the formula \( Q = mc\Delta T \), where \( Q \) is the heat supplied (1.8 \(\times\) 10^6 \ \(\text{J}\)), \( m \) is the mass of air (from step 2), \( c \) is the specific heat capacity (from step 3), and \( \Delta T \) is the temperature change. Rearrange the formula to solve for \( \Delta T \).

Substitute the known values for \( Q \), \( m \), and \( c \) into the equation from step 4 to calculate \( \Delta T \), the temperature rise in one hour.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Heat Transfer

Heat transfer refers to the process of thermal energy moving from one object or substance to another due to a temperature difference. In this scenario, the heater provides energy to the air in the room, which will increase the air's temperature. Understanding how heat is transferred and measured is crucial for calculating the resulting temperature change.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and number of moles of an ideal gas. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. This law helps in determining how the temperature of the air will change when heat is added.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius. For air, this value is essential to calculate how much the temperature will increase when a certain amount of heat is supplied. Knowing the specific heat capacity allows us to relate the energy supplied by the heater to the temperature change in the air.