Estimate (a) The number of molecules in the air in a room....

Estimate (a) The number of molecules in the air in a room. (b) Their energy, in joules or in ergs, per mole. (c) What quantity of heat (in joules or in ergs) must be added to warm one mole of air at 1 atm from $0^{\circ} \mathrm{C}$ to $20^{\circ} \mathrm{C}$ ? (d) What is the minimum energy that must be supplied to a refrigerator to cool 1 mole of air at 1 atm from $20^{\circ} \mathrm{C}$ to $18^{\circ} \mathrm{C}$ ? The refrigerator acts in a cyclic process and gives out heat at $40^{\circ} \mathrm{C}$.

Estimate
(a) The number of molecules in the air in a room.
(b) Their energy, in joules or in ergs, per mole.
(c) What quantity of heat (in joules or in ergs) must be added to warm one mole of air at 1 atm from $0^{\circ} \mathrm{C}$ to $20^{\circ} \mathrm{C}$ ?
(d) What is the minimum energy that must be supplied to a refrigerator to cool 1 mole of air at 1 atm from $20^{\circ} \mathrm{C}$ to $18^{\circ} \mathrm{C}$ ? The refrigerator acts in a cyclic process and gives out heat at $40^{\circ} \mathrm{C}$.

Key Concepts

Energy Unit Conversion

This concept covers the conversion between energy units such as joules and ergs, ensuring that all calculations are consistent and comparable. Mastery of unit conversion is necessary for accurately expressing energy values in the desired measurement system and for comparing magnitudes across different scales.

Thermodynamic Refrigeration and Carnot Efficiency

Refrigeration processes are governed by thermodynamic cycles, with the Carnot cycle providing the theoretical minimum work or energy required for a given heat transfer. Understanding the coefficient of performance (COP) for refrigerators, which relates the work input to the heat removed, is vital for determining the minimal energy needed to achieve a specified cooling under cyclic operation.

Kinetic Theory of Gases

Kinetic theory explains the behavior of gas molecules in terms of their energy, speed, and motion, linking temperature to the average kinetic energy of molecules. This theory allows one to estimate the energy per mole of gas molecules as a function of temperature, which is critical when calculating thermal energy content.

Mole and Avogadro's Number

This concept involves understanding that one mole of any substance contains approximately 6.02 x 10^23 entities, allowing for conversion between macroscopic quantities (like room volume) and microscopic counts (such as molecules). It is fundamental for estimating the number of molecules in a given sample and relates the measurable mass or volume to the number of constituent particles.

Heat Capacity and Calorimetry

Heat capacity is a measure of the amount of energy required to change a substance's temperature, and in the context of gases, it often involves using specific heat relationships at constant pressure. Calorimetry concepts help relate the quantity of heat added to the resultant temperature change, which is essential in determining the heat needed for warming the air.

Transcript

00:01 In this question, we are given the volume of the room.

00:08 There's a room, or volume is found to be 4 .2 meters times, times 3 meters, times 2 .5 meters.

00:19 So we want to find the information about the gas inside this room at atmospheric pressure and 20 degrees celsius.

00:30 So we want to find quantities like number of molecules, mass of the air, average kinetic energy of one air molecule, the room unit square speed, the internal energy of the air at 20 degrees celsius and 25 degrees celsius.

00:53 So part a for the number of molecules we're using the ideal gas equation.

00:59 We have pv equals to big n times kbt.

01:09 So n is pv divide by kbt.

01:14 The pressure is one atmosphere, so it's 1 .013 times 10 to the power 5.

01:20 Pascal, so the volume is 4 .2 times 3 times 2 .5.

01:26 Kb goes one constant.

01:31 And the temperature is 293.

01:36 So 20 degrees plus 2 .7.

01:38 You get 293 and n is calculated to be 7 .89 times 10 to the power 26 of molecules.

01:50 Okay, so this is the answer for part a.

01:55 Then in part b, you want to calculate the mass of this air.

02:00 Mass of air is equal to the number of molecules divided by the avocado's constant times the holmes.