Animated Solution for Physics - Properties of Solids and Liquids: A non-isotropic solid metal cube has coefficients of linear expansion as : $5\times 10^{-5}{/}^{\circ }\text{C}$ along the X-axis and $5\times 10^{-6}{/}^{\circ }\text{C}$ along the Y and the Z-axis. If the coefficient of volume expansion of the solid is $C\times 10^{-6}{/}^{\circ }\text{C}$, then the value of $C$ is …… .

Imagine you are holding a solid metal cube in your hands. In most textbook problems, when you heat this cube, it expands uniformly. It gets slightly taller, slightly wider, and slightly deeper, all by the exact same proportion. We call such materials isotropic, meaning their properties are identical in all directions.

But the universe is full of surprises, and the cube in our problem is a rebel. It is non-isotropic (or anisotropic). This means its atomic structure is arranged in such a way that it responds differently to heat depending on the direction you measure.

When we place this specific cube in an oven, it stretches significantly more along the X-axis than it does along the Y or Z axes. Specifically, its coefficient of linear expansion along the X-axis (${\alpha }_x$) is $5\times 10^{-5}{/}^{\circ }\text{C}$, while along the Y and Z axes (${\alpha }_y$ and ${\alpha }_z$), it is only $5\times 10^{-6}{/}^{\circ }\text{C}$. Notice that the expansion along the X-axis is ten times greater!

Our mission is to find the overall coefficient of volume expansion ($\gamma$) for this peculiar cube and determine the mystery constant $C$.

To understand how the volume changes, we must first define what the volume is. Let's set up a 3D coordinate system and align our cube with the X, Y, and Z axes. Let the lengths of the sides of the cube be $x$, $y$, and $z$.

The fundamental geometric truth for the volume $V$ of a cuboid is simply the product of its three dimensions:

This equation is our anchor. It tells us that any change in the overall volume $V$ must be a direct consequence of the changes in the individual side lengths $x$, $y$, and $z$.

Now, let's turn up the heat! As the temperature increases by a small amount $\Delta T$, the sides expand by $\Delta x$, $\Delta y$, and $\Delta z$. Consequently, the volume expands by $\Delta V$.

How do we mathematically link these small changes? We could use the product rule of differentiation, but there is a much more elegant trick: logarithmic differentiation.

By taking the natural logarithm ($\ln$) of both sides of our volume equation, we transform the multiplication into addition:

Now, we differentiate this entire equation. The derivative of $\ln (V)$ is $\frac{dV}{V}$. For small macroscopic changes, we can safely replace the infinitesimals ($d$) with deltas ($\Delta$). This yields a beautiful relationship:

This equation is profound. It states that the fractional change in volume is exactly equal to the sum of the fractional changes in each of the three linear dimensions.

We are almost there, but we need to introduce temperature into our math. Let's divide every single term in our equation by the change in temperature, $\Delta T$:

Look closely at these terms. Do they look familiar?

By definition, the fractional change in length per unit change in temperature ($\frac{\Delta L}{L\Delta T}$) is the coefficient of linear expansion, $\alpha$. Similarly, the fractional change in volume per unit change in temperature ($\frac{\Delta V}{V\Delta T}$) is the coefficient of volume expansion, $\gamma$.

Substituting these definitions into our equation, we arrive at the master equation for anisotropic thermal expansion:

This is a crucial takeaway. For any material, whether isotropic or anisotropic, the volume expansion coefficient is always the sum of the three orthogonal linear expansion coefficients.

Now comes the execution phase. We have our master equation, and we have the raw values from the problem. Let's substitute them in:

This is the exact moment where countless students make a fatal silly mistake. When you are under exam pressure, it is incredibly tempting to just look at the leading numbers and add $5+5+5=15$.

But you cannot directly add numbers in scientific notation if their exponents are different! It is mathematically equivalent to trying to add 5 kilometers to 5 meters without converting the units first.

We must force all the terms to have the exact same power of 10. Let's target $10^{-6}$ as our common base.

How do we convert $5\times 10^{-5}$? We can multiply the coefficient by 10 and divide the power of 10 by 10.

Now, let's rewrite our substitution with this uniform power:

With the powers of 10 perfectly aligned, the arithmetic becomes trivial. We can factor out the $10^{-6}$ from every term:

Adding the numbers inside the parenthesis gives us:

We have successfully calculated the coefficient of volume expansion for our rebel cube.

The final step is to answer the specific question asked. The problem states that the coefficient of volume expansion is given by the expression $C\times 10^{-6}{/}^{\circ }\text{C}$.

By directly comparing our calculated result with the given expression:

It is undeniably clear that the mystery constant $C$ is exactly 60.

Final Answer: The value of $C$ is 60.