Answer :
Answer:
793 kJ
Explanation:
Given that :
[tex]T_1 = 1760^0C\\\\T_2 = 2060^0C\\\\ c_s = 44.8 \ J/g \ ^0C[/tex]
Then; the heat needed to convert solid from [tex]1760^0C \ to \ 1860^0C[/tex] is calculated as:
[tex]Q_1 = mc_s \delta T[/tex]
[tex]Q_1 = mc_s (T_f - T_i)[/tex]
[tex]Q_1 = 159.6*44.8(1860-1760)^0C \\ \\ Q_1 = 715008 \ J[/tex]
Also;
[tex]\delta H_f = 20.5 \ kJ/mol = 20500 \ J/mol \\ \\ Molar \ mass \ of \ Cr = 52.0 g/mol \\ \\ number \ of \ moles\ of \ Cr = \frac{mass \ of \ Cr}{molar \ mass} \\ \\ = \frac{159.6}{52.0}\\\\= 3.069 \ mol[/tex]
Now; the heat required to convert solid to liquid at [tex]1860^0C[/tex] is;
[tex]Q_2 = n* \delta H_f[/tex]
= 3.069 × 20500
= 62914.5 J
Also ; given that :
[tex]c__l}} = 0.94 \ J/g \ ^0C \\ \\[/tex]
Then the heat needed to convert liquid from [tex]1860^0C[/tex]to [tex]2060^0C[/tex] is;
[tex]Q_3 = m*c__l} (T_f-T_l)[/tex]
[tex]Q_3 = 159.6*0.94*(2060-1760)[/tex]
[tex]Q_3 = 15002.4 \ J[/tex]
∴ The total heat required = [tex]Q_1 + Q_2 + Q_3[/tex]
= (715008 + 62914.5 + 15002.4) J
= 792924.9 J
= 793 kJ
Teking into accoun the definition of calorimetry, sensible heat and latent heat, the amount of heat required is 808 kJ.
Calorimetry is the measurement and calculation of the amounts of heat exchanged by a body or a system.
Sensible heat is defined as the amount of heat that a body absorbs or releases without any changes in its physical state (phase change).
Latent heat is defined as the energy required by a quantity of substance to change state.
When this change consists of changing from a solid to a liquid phase, it is called heat of fusion and when the change occurs from a liquid to a gaseous state, it is called heat of vaporization.
- 1760 °C to 1860 °C
In firts place, you know that the melting point is 1860°C. So, first of all you must increase the temperature from 1760 ° C (in solid state) to 1860 ° C, in order to supply heat without changing state (sensible heat).
The amount of heat a body receives or transmits is determined by:
Q = c× m× ΔT
where Q is the heat exchanged by a body of mass m, made up of a specific heat substance c and where ΔT is the temperature variation.
In this case, you know:
- c(solid)= 44.8 [tex]\frac{J}{gC}[/tex]
- m= 159.4 g
- ΔT= Tfinal - Tinitial= 1860 °C - 1760 °C= 100 °C
Replacing:
Q1= 44.8 [tex]\frac{J}{gC}[/tex]× 159.6 g× 100 °C
Solving:
Q1=715,008 J= 715.008 kJ
- Change of state
The heat Q that is necessary to provide for a mass m of a certain substance to change phase is equal to
Q = m×L
where L is called the latent heat of the substance and depends on the type of phase change.
In this case, you know:
- n= 159.6 grams×[tex]\frac{1 mole}{52 grams}[/tex]= 3.069 moles, where 52[tex]\frac{gr}{mole}[/tex] is the molar mass of Cr, that is, the amount of mass that a substance contains in one mole.
- ΔHfus= 20.5 [tex]\frac{kJ}{mol}[/tex]
Replacing:
Q2= 3.069 moles×20.5 [tex]\frac{kJ}{mol}[/tex]
Solving:
Q2=62.9145 J
- 1860 °C to 2060 °C
Similar to sensible heat previously calculated, you know:
- c(liquid)= 0.94 [tex]\frac{J}{gC}[/tex]
- m= 159.4 g
- ΔT= Tfinal - Tinitial= 2060 °C - 1860 °C= 200 °C
Replacing:
Q3= 0.94 [tex]\frac{J}{gC}[/tex]× 159.6 g× 200 °C
Solving:
Q3=30,004.8 J= 30.0048 kJ
- Total heat required
The total heat required is calculated as:
Total heat required= 715.008 kJ + 62.9145 kJ + 30.0048 kJ
Total heat required= 807.9273 kJ ≅ 808 kJ
In summary, the amount of heat required is 808 kJ.
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