Show that the bulk modulus (Section 12–5) for an ideal gas held at constant temperature is B = P, where P is the pressure.

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The bulk modulus (B) is defined as the negative ratio of the change in pressure (ΔP) to the fractional change in volume (ΔV/V). Mathematically, B = -ΔP / (ΔV/V).

For an ideal gas at constant temperature (isothermal process), the ideal gas law applies: PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is the temperature.

Rearrange the ideal gas law to express pressure as P = nRT / V. Since temperature (T) is constant, nRT is also constant.

Differentiate P = nRT / V with respect to V to find the relationship between pressure and volume. This gives dP/dV = -nRT / V². Substitute this into the bulk modulus formula.

Using B = -ΔP / (ΔV/V) and substituting dP/dV = -nRT / V², simplify the expression to show that B = P. This demonstrates that the bulk modulus for an ideal gas at constant temperature equals the pressure.

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주요 개념

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Bulk Modulus

The bulk modulus (B) is a measure of a material's resistance to uniform compression. It is defined as the ratio of the change in pressure to the fractional change in volume. For an ideal gas, the bulk modulus indicates how much pressure is required to compress the gas by a certain volume, reflecting its compressibility.

Ideal Gas Law

The ideal gas law is a fundamental equation in thermodynamics that relates pressure (P), volume (V), temperature (T), and the number of moles (n) of a gas. It is expressed as PV = nRT, where R is the ideal gas constant. This law assumes that gas particles do not interact and occupy no volume, making it applicable under certain conditions.

Isothermal Process

An isothermal process is a thermodynamic process that occurs at a constant temperature. For an ideal gas, this means that any change in pressure or volume must occur in such a way that the product of pressure and volume remains constant. In this context, the relationship between pressure and volume during compression or expansion is crucial for deriving the bulk modulus.