Sure, let's go through the problem step-by-step.
### Problem Statement Recap
We need to calculate the amount of heat required to heat a cube of iron from [tex]\(25.0^{\circ} C\)[/tex] to [tex]\(49.0^{\circ} C\)[/tex]. We'll use the specific heat capacity of iron [tex]\(C_p\)[/tex], the mass of the cube, and the initial and final temperatures to find the heat required using the formula:
[tex]\[ q = m \times C_p \times \Delta T \][/tex]
### Given Values
1. Specific heat capacity, [tex]\(C_p = 0.450 \, \text{J/(g} \cdot {}^{\circ}\text{C)}\)[/tex]
2. Mass of the iron cube, [tex]\(m = 55.8 \, \text{g}\)[/tex]
3. Initial temperature, [tex]\(T_i = 25.0^{\circ}C\)[/tex]
4. Final temperature, [tex]\(T_f = 49.0^{\circ}C\)[/tex]
### Steps to Solve
1. Calculate the temperature change ([tex]\(\Delta T\)[/tex]):
[tex]\[ \Delta T = T_f - T_i \][/tex]
Substituting the given values:
[tex]\[ \Delta T = 49.0^{\circ} C - 25.0^{\circ} C \][/tex]
[tex]\[ \Delta T = 24.0^{\circ} C \][/tex]
2. Plug the values into the formula:
[tex]\[ q = m \times C_p \times \Delta T \][/tex]
Using the given values:
[tex]\[ q = 55.8 \, \text{g} \times 0.450 \, \text{J/(g} \cdot {}^{\circ}\text{C)} \times 24.0^{\circ}C \][/tex]
3. Calculate the heat required (q):
[tex]\[ q = 55.8 \times 0.450 \times 24.0 \][/tex]
### Numerical Calculation
[tex]\[ q = 55.8 \times 0.450 = 25.11 \][/tex]
[tex]\[ q = 25.11 \times 24.0 = 602.64 \, \text{J} \][/tex]
### Conclusion
The heat required to heat the iron cube from [tex]\(25.0^{\circ}C\)[/tex] to [tex]\(49.0^{\circ}C\)[/tex] is:
[tex]\[ \boxed{602.64 \, \text{J}} \][/tex]
Rounding to three significant figures, the heat required is:
[tex]\[ \boxed{603 \, \text{J}} \][/tex]
So, [tex]\( 603\, \text{J} \)[/tex] of heat is needed to heat the iron cube from [tex]\( 25.0^{\circ} C \)[/tex] to [tex]\( 49.0^{\circ} C \)[/tex].
