To understand how the coefficient of cubical expansion varies with temperature for an ideal gas, we need to delve into the relationships between volume, temperature, and pressure. The coefficient of cubical expansion, often denoted as β, describes how the volume of a substance changes with temperature. For an ideal gas, this relationship is not as straightforward as it might seem at first glance.

Understanding the Coefficient of Cubical Expansion

The coefficient of cubical expansion (β) is defined as the fractional change in volume per degree change in temperature. For solids and liquids, this coefficient is typically considered constant over a small temperature range. However, for gases, especially ideal gases, the situation is different due to their compressibility and the nature of their molecular interactions.

Ideal Gas Law and Volume Change

According to the ideal gas law, the relationship between pressure (P), volume (V), and temperature (T) for an ideal gas is given by:

• P * V = n * R * T

Where:

• n = number of moles of gas
• R = universal gas constant

When the volume of the gas is increased while keeping the pressure constant, we can express the change in volume (ΔV) as a function of temperature. If the initial volume is V and the initial temperature is T, and the temperature increases to T + ΔT, the new volume (V') can be expressed as:

• V' = V * (1 + β * ΔT)

Variation of the Coefficient with Temperature

For an ideal gas, the coefficient of cubical expansion is not a constant; it actually varies with temperature. The expression for the coefficient of cubical expansion for an ideal gas can be derived from the ideal gas law. When we keep pressure constant and increase temperature, the volume expands according to:

• β = (1/V) * (dV/dT) = (nR/P) = (1/T)

This shows that the coefficient of cubical expansion for an ideal gas is directly proportional to the temperature (T). As the temperature increases, the coefficient β increases as well, which means that the gas expands more for each degree of temperature increase at higher temperatures.

Practical Implications

In practical terms, this means that if you were to heat a balloon filled with an ideal gas, the balloon would expand more significantly at higher temperatures than at lower temperatures. This is crucial in applications such as engines, where gas expansion plays a vital role in efficiency and performance.

In summary, while the coefficient of cubical expansion for solids and liquids can often be treated as constant, for ideal gases, it varies with temperature, increasing as the temperature rises. This relationship is essential for understanding gas behavior in various scientific and engineering contexts.