Problem 114 - Chemistry

Question

The element gallium (Ga) has the second largest liquid range of any element, melting at \(29.78{ }^{\circ} \mathrm{C}\) and boiling at \(2204{ }^{\circ} \mathrm{C}\) at atmospheric pressure. (a) What is the density of gallium in \(\mathrm{g} / \mathrm{cm}^{3}\) at \(25^{\circ} \mathrm{C}\) if a 1 in. cube has a mass of \(0.2133 \mathrm{lb} ?\) (b) Assume that you construct a thermometer using gallium as the fluid instead of mercury, and that you define the melting point of gallium as \(0{ }^{\circ} \mathrm{G}\) and the boiling point of gallium as \(1000^{\circ} \mathrm{G}\). What is the melting point of sodium chloride \(\left(801{ }^{\circ} \mathrm{C}\right)\) on the gallium scale?

Step-by-Step Solution

Answer

(a) Density is about 5.902 g/cm³. (b) Melting point of NaCl is about 354.69 °G on the Ga scale.

Step 1: Convert Mass to Grams

Start by converting the mass of the gallium cube from pounds to grams. Since the mass of the cube is given as 0.2133 lb, we use the conversion factor: 1 lb = 453.59237 g.So, the mass in grams is:\[0.2133 \text{ lb} \times 453.59237 \frac{\text{g}}{\text{lb}} = 96.75 \text{ g}\]

Step 2: Convert Volume to Cubic Centimeters

The volume of the cube is numerically found by cubing the side length. Since it's a 1-inch cube, 1 in = 2.54 cm, thus:\[V = (2.54 \text{ cm})^3 = 16.387064 \text{ cm}^3\]

Step 3: Calculate Density

Density \(\rho\) is mass divided by volume. Using the results from the previous steps:\[\rho = \frac{\text{mass}}{\text{volume}} = \frac{96.75 \text{ g}}{16.387064 \text{ cm}^3} \approx 5.902 \text{ g/cm}^3\]

Step 4: Establish Gallium Temperature Scale

In this step, we define the gallium temperature scale where 0 G corresponds to \(29.78^{\circ} \text{C}\) (melting point) and 1000 G to \(2204^{\circ} \text{C}\) (boiling point).

Step 5: Calculate Scale Factor and Offset

To convert Celsius to Gallium degrees, first calculate the range:\[\Delta T_C = 2204^{\circ}C - 29.78^{\circ}C = 2174.22^{\circ}C\]Correspondingly, for Gallium scale:\[\Delta T_G = 1000^{\circ}G - 0^{\circ}G = 1000^{\circ}G\]The scale factor \(k\) is:\[k = \frac{2174.22^{\circ}C}{1000^{\circ}G} \approx 2.17422 \frac{^{\circ}C}{^{\circ}G}\]

Step 6: Convert Melting Point of NaCl

To find the melting point of NaCl in Gallium degrees, we translate from Celsius to Gallium scale:First, calculate the difference from gallium's melting point:\[T_C - 29.78^{\circ}C = 801^{\circ}C - 29.78^{\circ}C = 771.22^{\circ}C\]Convert this difference using the scale factor:\[T_G = \frac{771.22^{\circ}C}{2.17422} \approx 354.69^{\circ}G\]Thus, the melting point of sodium chloride on the Gallium scale is approximately 354.69 \(^{\circ}G\).

Key Concepts

Gallium Temperature Scale

When discussing temperature scales, it is often helpful to refer to both well-known and lesser-known scales. In the case of gallium, a custom temperature scale can be developed due to its unique melting and boiling points. Starting at the melting point, gallium solidifies at 29.78°C, which becomes 0°G on this gallium scale. The boiling point, at 2204°C, is defined as 1000°G on this scale.

To understand the temperature conversion between Celsius and this special gallium scale, we calculate the difference in temperature between the solid and vapor states of gallium. This equates to a range of 2174.22°C. Meanwhile, the equivalent range on the gallium scale is 1000°G. With this distinction, we derive a scale factor to help convert between Celsius and Gallium degrees, offering an intriguing connection between standard and specialized measurements.

Cubic Centimeter Conversion

Whenever dealing with various measurement systems, conversions become crucial. In our exercise, converting units from inches to centimeters is a necessary step to determine the volume of a cubic object in cubic centimeters.

A cube measured in inches can be transferred to the metric system using the conversion 1 inch = 2.54 cm. Therefore, a 1-inch dimension corresponds to a side length of 2.54 cm. To find the volume, cube the side length: \[ V = (2.54 \text{ cm})^3 = 16.387064 \text{ cm}^3 \] This conversion process is vital, as it enables us to work within compatible units for subsequent calculations, like computing density in terms of grams per cubic centimeter.

Celsius to Gallium Conversion

To convert temperatures from the Celsius scale to the Gallium scale, first, we identify the set points for both scales. Gallium's melting and boiling points have been defined as 0°G and 1000°G, equivalent to 29.78°C and 2204°C, respectively.

The conversion requires applying a scale factor which we determined as approximately 2.17422 °C per Gallium degree. Thus, to convert a specific temperature, such as the melting point of sodium chloride (801°C), to the Gallium scale:

• First, find the offset from gallium’s melting point:\[ 801^{\circ}C - 29.78^{\circ}C = 771.22^{\circ}C \]
• Use the scale factor to convert this offset:\[ T_G = \frac{771.22^{\circ}C}{2.17422} \approx 354.69^{\circ}G \]

These steps provide a structured method to shift measurements from conventional temperature scales to a specialized format, helping to apply gallium-centric calculations and experimental setups.