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KIM [24]
3 years ago
15

A 62.6-gram piece of heated limestone is placed into 75.0 grams of water at 23.1°C. The limestone and the water come to a final

temperature of 51.9°C. The specific heat capacity of water is 4.186 joules/gram degree Celsius, and the specific heat capacity of limestone is 0.921 joules/gram degree Celsius. What was the initial temperature of the limestone? Express your answer to three significant figures.
Chemistry
2 answers:
KATRIN_1 [288]3 years ago
8 0

Answer : The initial temperature of the limestone is 1.05\times 10^{2}^oC.

Solution : Given,

Mass of limestone = 62.6 g

Mass of water = 75 g

Final temperature of limestone = 51.9^oC

Final temperature of water = 51.9^oC

Initial temperature of water = 23.1^oC

The specific heat capacity of limestone = 0.921J/g^oC

The specific heat capacity of water = 4.186J/g^oC

The formula used :

q = m × c × ΔT

where,

q = heat required

m = mass of an element

c = heat capacity

ΔT = change in temperature

According to this, the energy as heat lost by the system is equal to the gained by the surroundings.

Now the above formula converted and we get

q_{system}=-q_{surrounding}

m_{system}\times c_{system}\times (T_{final}-T_{initial})_{system}= -[m_{surrounding}\times c_{surrounding}\times (T_{final}-T_{initial})_{surrounding}]

m_{limestone}\times c_{limestone}\times (T_{final}-T_{initial})_{limestone}= -[m_{water}\times c_{water}\times (T_{final}-T_{initial})_{water}]

Now put all the given values in this formula, we get

62.6g\times 0.921J/g^{0}C \times (51.9^{0}C-T_{\text{ initial limestone}})= -[75g\times 4.186J/g^{0}C \times (51.9^{0}C-23.1^oC)]

By rearranging the terms, we get  the value of initial temperature of limestone.

T_{\text{ initial limestone}}=104.926^{0}C=1.05\times 10^{2}^oC

Therefore, the initial temperature of the limestone is 1.05\times 10^{2}^oC.




Lemur [1.5K]3 years ago
5 0

m₁ = mass of water = 75 g

T₁ = initial temperature of water = 23.1 °C

c₁ = specific heat of water = 4.186 J/g°C


m₂ = mass of limestone = 62.6 g

T₂ = initial temperature of limestone = ?

c₂ = specific heat of limestone = 0.921 J/g°C


T = equilibrium temperature = 51.9 °C

using conservation of heat

Heat lost by limestone = heat gained by water

m₂c₂(T₂ - T) = m₁c₁(T - T₁)

inserting the values

(62.6) (0.921) (T₂ - 51.9) = (75) (4.186) (51.9 - 23.1)

T₂ = 208.73 °C

in three significant  figures

T₂ = 209 °C

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Approximately 4.92.

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Initial quantity of \rm NH_3:

\begin{aligned} n({\rm NH_3}) &= c({\rm NH_3}) \cdot V({\rm NH_3}) \\ &= 0.3733\; \rm mol \cdot L^{-1} \times 0.1900\; \rm L \\ &\approx 0.154375\; \rm mol\end{aligned}.

Ammonia \rm NH_3 reacts with hydrochloric \rm HCl acid at a one-to-one ratio:

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Hence, approximately n({\rm HCl}) = 0.154375\; \rm mol of \rm HCl\! molecules would be required to exactly react with the \rm NH_3\! in the original solution and hence reach the equivalence point of this titration.

Calculate the volume of that 0.3733\; \rm mol \cdot L^{-1} \rm HCl solution required for reaching the equivalence point of this titration:

\begin{aligned}V({\rm HCl}) &= \frac{n({\rm HCl})}{c({\rm HCl})} \\ &\approx \frac{0.154375\; \rm mol}{0.3733\; \rm mol \cdot L^{-1}} \approx 0.413541\; \rm L\end{aligned}.

Hence, by the assumption stated in the question, the volume of the solution at the equivalence point would be approximately 0.413541\; \rm L + 0.1900\; \rm L \approx 0.6035\; \rm L.

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\begin{aligned}c({\rm {NH_4}^{+}}) &= \frac{n({\rm {NH_4}^{+}})}{V({\rm {NH_4}^{+}})}\\ &\approx \frac{0.154375\; \rm mol}{0.6035\; \rm L} \approx 0.255782\; \rm mol \cdot L^{-1}\end{aligned}.

However, because \rm NH_3 \cdot H_2O is a weak base, its conjugate \rm {NH_4}^{+} would be a weak base.

\begin{aligned}pK_{\rm a}({{\rm NH_4}}^{+}) &= pK_{\rm w} - pK_{\rm b}({\rm NH_3})\\ &\approx 13.99 - 4.75 = 9.25\end{aligned}.

Hence, the following reversible reaction would be take place in the solution at the equivalence point:

\rm {NH_4}^{+} \rightleftharpoons NH_3 + H^{+}.

Let x\; \rm mol \cdot L^{-1} be the increase in the concentration of \rm H^{+} in this solution because of this reversible reaction. (Notice that x \ge 0.) Construct the following \text{RICE} table:

\begin{array}{c|ccccc} \textbf{R}& \rm {\rm NH_4}^{+} & \rightleftharpoons & {\rm NH_3}& + & {\rm H}^{+}\\ \textbf{I} & 0.255782 \; \rm M \\ \textbf{C} & -x \;\rm M & & + x\;\rm M & & + x\; \rm M \\ \textbf{E} & (0.255782 - x)\; \rm M & & x\; \rm M & & x\; \rm M\end{array}.

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