A very thin layer of gold plating was placed on a metal plate that measured 25.22 by 13.22 cm. The gold plating increased the mass of the plate by 0.00512g. Calculate the thickness of the plating. The density of gold is 19.32 g/cm3.

To find the thickness of the gold plating, we can use the relationship between mass, volume, and density.

The formula for density is:

\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}}
\]

Rearranging the formula to solve for volume:

\[
\text{Volume} = \frac{\text{Mass}}{\text{Density}}
\]

### Step 1: Calculate the volume of the gold plating

Given:
- The mass of the gold plating is \( 0.00512 \, \text{g} \),
- The density of gold is \( 19.32 \, \text{g/cm}^3 \).

Now, the volume of the gold plating is:

\[
V = \frac{0.00512 \, \text{g}}{19.32 \, \text{g/cm}^3}
\]

Let's calculate the volume.

\[
V \approx 0.000265 \, \text{cm}^3
\]

### Step 2: Relating volume to the surface area and thickness

The volume of the gold layer can also be written as:

\[
V = \text{Area of the plate} \times \text{Thickness of the plating}
\]

The area of the plate is:

\[
\text{Area} = 25.22 \, \text{cm} \times 13.22 \, \text{cm} = 333.9004 \, \text{cm}^2
\]

Let \( t \) be the thickness of the gold plating. Thus, the volume is:

\[
V = 333.9004 \, \text{cm}^2 \times t
\]

Using the calculated volume from Step 1:

\[
0.000265 \, \text{cm}^3 = 333.9004 \, \text{cm}^2 \times t
\]

### Step 3: Solve for \( t \)

\[
t = \frac{0.000265 \, \text{cm}^3}{333.9004 \, \text{cm}^2}
\]

\[
t \approx 7.94 \times 10^{-7} \, \text{cm}
\]

### Final Answer:
The thickness of the gold plating is approximately \( 7.94 \times 10^{-7} \, \text{cm} \).

Would you like more details on any part of this process?

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Here are 5 related questions to deepen understanding:
1. What is the significance of the density of a material in determining thickness?
2. How would the result change if the dimensions of the plate were different but the mass increase was the same?
3. What other factors could affect the uniformity of gold plating on a surface?
4. How can the accuracy of this method of determining thickness be improved?
5. What are practical applications of determining the thickness of coatings?

**Tip:** Always ensure that your units are consistent when applying formulas in physics and chemistry.

Learn how to calculate the thickness of a thin gold plating layer on a metal plate with dimensions 25.22 by 13.22 cm, and a mass increase of 0.00512g. This problem involves key concepts such as volume, density, and algebraic manipulation, making it suitable for high school students in Grades 10-12.

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