Compute the density for iron at \(700^{\circ} \mathrm{C}\), given that its room- temperature density is \(7.870 \mathrm{~g} / \mathrm{cm}^{3} .\) Assume that the volume coefficient of thermal expansion, \(\alpha_{v}\), is equal to \(3 \alpha_{l}\).

When we talk about the density of a material, we're referring to how much mass there is in a given volume. It's a fundamental concept in physics and engineering that helps us understand how substances will behave under different conditions. The formula for density is quite simple:
\[ \rho = \frac{m}{V} \]
In this formula, \( \rho \) stands for density, \( m \) for mass, and \( V \) for volume. But here’s the catch—volume can change when the temperature changes. This is where thermal expansion comes into play. Even if the mass of a material like iron remains constant, the volume can increase if the temperature rises, causing the density to decrease.
In our exercise, we focus on how the density of iron changes when it’s heated to \(700^\circ \mathrm{C}\). To figure out the new density, we first need to understand how much the volume increases due to thermal expansion. Once we have the new volume, the mass (which stays the same regardless of temperature) can be divided by the new volume to find the updated density.

The volume coefficient of thermal expansion, usually represented by \(\alpha_{v}\), is crucial in understanding how the volume of materials changes with temperature. This coefficient shows us the rate at which a material's volume is expected to expand per degree change in temperature. The formula that describes this is:
\[ dV = \alpha_{v} V dT \]
The change in volume \(dV\) is proportional to both the original volume \(V\) and the change in temperature \(dT\), with the volume expansion coefficient \(\alpha_{v}\) serving as the constant of proportionality. Since our problem states that the volume expansion coefficient is three times the linear coefficient \(\alpha_{l}\), we use this relationship to express the volume change in terms of \(\alpha_{l}\). This understanding allows us to predict the new volume after a temperature increase and, subsequently, the new density of the material as it expands.

The linear expansion coefficient \(\alpha_{l}\) gives us a glimpse into how the physical dimensions of a material change linearly with temperature. Specifically, it indicates how much a unit length of the material will expand (or contract) per degree change in temperature.
In our scenario, knowing the linear expansion coefficient is key to determining the change in volume since the volume coefficient \(\alpha_{v}\) is given to be three times a linear coefficient \(3\alpha_{l}\). The relationship between them allows us to predict how a material’s volume will change when subjected to a temperature change. For isotropic materials, which expand uniformly in all directions, this connection is straightforward because the volume expansion is basically the linear expansion happening in three dimensions. It’s essential to grasp that without knowing the value of \(\alpha_{l}\), we can't accurately calculate the density of iron at higher temperatures, as it directly influences the volume change.