Modelling gases - the kinetic model - class-XII

1. Nitrogen $-78$%
   2) Oxygen $-21$%
   3) Inert Nobel gases $-0.9$%
   4) Several trace gases $-0.1$%
   The above list the different number of gases in descending order. As per the list clearly Nitrogen is the gas present in most air.

Heating up a substance excites the molecules of that substance making  the molecules moves vigorously thus giving velocity to the molecules and we know kinetic energy of a body is by the virtue of its motion.       

Kinetic energy K = (3/2)nRT. For 20g of oxygen (molar mass 32g/mol), n = 20/32 = 0.625 mol. T = 47 + 273 = 320 K. K = (3/2) * 0.625 * 8.3 * 320 = 2490 J.

An ideal gas is a gas which satisfying the assumptions of the kinetic energy.

Statement 1 is true as internal energy of an ideal gas depends only on temperature. Statement 2 is true because at constant pressure, some heat is used for work (expansion), leaving less for temperature rise compared to constant volume. However, Statement 2 does not explain Statement 1.

According to the Stefan-Boltzmann law, radiated energy per unit time is E = e * sigma * A * T^4. Substituting the given values (T = 727 + 273 = 1000 K, A = 5 cm^2 = 5 * 10^-4 m^2, E = 300 J / 60 s = 5 W), solving for emissivity e yields 0.18.

From the kinetic theory of gases, the pressure exerted by an ideal gas is given by P = (1/3) * rho * v_rms^2, where rho is density (m*n) and v_rms^2 is the mean square speed (V^2_av).

Speed of sound v = sqrt(gamma * P / rho). Since P/rho is constant for an ideal gas at constant temperature, speed is independent of pressure. The change in speed depends only on temperature change: dv/dT = v / (2T). For air at 273K, v = 332 m/s. dv = (332 / (2 * 273)) * (-1) = -0.61 m/s = -61 cm/s. The magnitude is 61 cm/s.

Here $P$ and $a/{ V }^{ 2 }$ are added that means they have same dimensions.

Options:
(A): One can not say velocity of molecule decreases  in each collision.
(B): The pressure exerted by a diatomic gas is proportional to rms speed, not mean speed.
(C): Since temp is constant, KE will be constant.
(D): Mean translational KE is proportional to temp, hence will increase with increase in absolute temp.

The mean free path lambda = 1 / (sqrt(2) * pi * d^2 * n). Since n = P / (kT), lambda = kT / (sqrt(2) * pi * d^2 * P). Thus, lambda is directly proportional to temperature T.

Pressure is force per unit area

Solids expand on heating because the potential energy curve of interatomic bonds is asymmetric. As atoms vibrate with higher energy at elevated temperatures, their average interatomic distance increases, leading to thermal expansion.

Pressure is created by the atom of the gas when the collide with the container. during collision the atoms apply force on the container wall. This force per unit area is the pressure of the gas.

Average kinetic energy of an ideal gas is directly proportional to its absolute temperature in Kelvin, so E is proportional to T. Converting the given Celsius temperatures to Kelvin, T1 = 27 + 273 = 300K and T2 = 327 + 273 = 600K. Since the temperature doubles from 300K to 600K, the average kinetic energy also doubles to 2E.

Of the solar radiation reaching the earths atmosphere
36% is reflected back into space
18% is absorbed by the earths atmosphere
This implies that only 46% reaches the earths surface
If absorbed by a black body, expect that the maximum temperature of this body is given by
where $I _{sc} = 1353 Wm^{-2}$