To calculate the amount of magnesium (Mg) required to increase the temperature of a certain volume of water, we will use the specific heat capacity of water and the formula for heat transfer. The specific heat capacity of water is approximately \(4.18 , \text{J/g}^\circ\text{C}\).
Step 1: Calculate the Mass of Water
Given the density of water is \(0.997 , \text{g/mL}\) and the volume is \(72 , \text{mL}\):
\[ \text{mass of water} = \text{volume} \times \text{density} = 72 , \text{mL} \times 0.997 , \text{g/mL} = 71.784 , \text{g} \approx 71.79 , \text{g} \]
Step 2: Calculate the Heat Absorbed by Water
To find the energy needed to increase the temperature of the water, we need to know the temperature change (\(\Delta T\)). Assuming that we want to raise the temperature by \( \Delta T \) degrees Celsius (you would need to specify this value if it's not given; let's assume the desired increase in temperature is \( 1^\circ\text{C} \) for this example):
\[ Q = m \cdot c \cdot \Delta T \]
Where:
- \(Q\) is the heat (in Joules),
- \(m\) is the mass of the water (71.79 g),
- \(c\) is the specific heat capacity of water (\(4.18 , \text{J/g}^\circ\text{C}\)),
- \(\Delta T\) is the temperature increase (\(1^\circ\text{C}\)).
\[ Q = 71.79 , \text{g} \times 4.18 , \text{J/g}^\circ\text{C} \times 1 ,^\circ\text{C} = 299.5582 , \text{J} \approx 299.56 , \text{J} \]
Step 3: Calculate the Amount of Magnesium Required
The reaction of magnesium with oxygen produces energy. A common reaction for magnesium is:
\[ 2 \text{Mg} + \text{O}_2 \rightarrow 2 \text{MgO} + \text{Energy} \]
The heat released can vary depending on the reaction specifics and conditions, but a commonly used average is about \( 1200 , \text{kJ/mol} \) of Mg reacted. Therefore, for 1 mole of Mg:
\[ \frac{1200 \times 10^3 , \text{J}}{1 , \text{mol}} = 1200 , \text{kJ} \]
The molar mass of magnesium (Mg) is approximately \( 24.31 , \text{g/mol} \).
Step 4: Calculate Moles of Magnesium Required
To find the moles required to produce the calculated energy:
\[ \text{moles of Mg} = \frac{Q}{\text{Energy released per mole of Mg}} = \frac{299.56 , \text{J}}{1200 \times 10^3 , \text{J/mol}} = 2.49633 \times 10^{-4} , \text{mol} \]
Step 5: Convert Moles to Grams
Using the molar mass to convert moles to grams:
\[ \text{mass of Mg} = \text{moles of Mg} \times \text{molar mass of Mg} = 2.49633 \times 10^{-4} , \text{mol} \times 24.31 , \text{g/mol} \approx 0.00608 , \text{g} \]
Final Answer
Express the answer in grams to two significant figures:
\[ \text{mass of Mg} \approx 0.0061 , \text{g} \]
Therefore, the mass of magnesium needed is approximately 0.0061 g.
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