One kg of adiatomic gas is at a pressure of 5 ×10^5( N / m^2) The density of the gas is {(5 kg)

step 1 Hello friends in the question it is given that 1k of a diatomic gas is at a pressure 5 into 10 t

One kg of adiatomic gas is at a pressure of 5 ×10^5( N /...

One $\mathrm{kg}$ of adiatomic gas is at a pressure of $5 \times 10^{5}\left(\mathrm{~N} / \mathrm{m}^{2}\right)$ The density of the gas is $\left\{(5 \mathrm{~kg}) / \mathrm{m}^{3}\right\}$ what is the energy of the gas due to its thermal motion ? (A) $2.5 \times 10^{5} \mathrm{~J}$ (B) $3.5 \times 10^{5} \mathrm{~J}$ (C) $4.5 \times 10^{5} \mathrm{~J}$ (D) $1.5 \times 10^{5} \mathrm{~J}$

One $\mathrm{kg}$ of adiatomic gas is at a pressure of $5 \times 10^{5}\left(\mathrm{~N} / \mathrm{m}^{2}\right)$ The density of the gas is $\left\{(5 \mathrm{~kg}) / \mathrm{m}^{3}\right\}$ what is the energy of the gas due to its thermal motion ?
(A) $2.5 \times 10^{5} \mathrm{~J}$
(B) $3.5 \times 10^{5} \mathrm{~J}$
(C) $4.5 \times 10^{5} \mathrm{~J}$
(D) $1.5 \times 10^{5} \mathrm{~J}$

Key Concepts

Ideal Gas Law

The ideal gas law (p = ?R_specificT or pV = nRT) establishes a relationship between pressure, volume, temperature, and density of a gas. It is foundational for analyzing the thermodynamic properties of gases, as it allows one to calculate unknown quantities when others are given. In problems concerning energy due to thermal motion, the ideal gas law is often used to determine temperature or volume from given pressure and density values, facilitating the calculation of internal energy.

Density and Volume Relationship

The relationship between density and volume is a straightforward but essential concept in thermodynamics. Given the mass of a gas, its volume can be determined by dividing the mass by the density (V = mass/?). This calculation is often a necessary step when using expressions that involve the product of pressure and volume (pV) to compute quantities such as the internal energy of the gas.

Internal Energy of an Ideal Gas

The internal energy of an ideal gas is the total energy contained in the microscopic motions of its molecules. It depends solely on temperature for an ideal gas, and is often expressed using the degrees of freedom of the molecules. In many cases, it can be calculated using formulas such as U = (f/2) nRT or in terms of pressure and volume as U = pV/(? - 1), where ? (gamma) is the specific heat ratio. This concept is central to understanding how the microscopic thermal motions contribute to the energy stored in a gas.

Specific Heat Ratio (?)

The specific heat ratio, ?, is the ratio of the specific heat at constant pressure (c?) to the specific heat at constant volume (c?). It plays a critical role in analyzing thermodynamic processes involving gases because it relates changes in pressure, volume, and temperature. For diatomic gases at moderate temperatures, ? is usually around 1.4. This value is used in formulas that connect pressure and volume to the internal energy of the gas, thereby indicating how energy is partitioned among the degrees of freedom.

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