Afegint el terme d'escalfament del gel:
Energia a dissipar per la beguda:
\[ Q_{\text{beguda}} = m_{\text{beguda}} \cdot c_{\text{beguda}} \cdot \Delta T \] \[ Q_{\text{beguda}} = 0.33 \cdot 4186 \cdot (25 - 5) = 27,628.8 \, \text{J} \]Energia absorbida pel gel:
\[ Q_{\text{gel}} = m_{\text{gel}} \cdot c_{\text{gel}} \cdot \Delta T + m_{\text{gel}} \cdot L_f + m_{\text{gel}} \cdot c_{\text{aigua}} \cdot 5 \] Amb: - \( c_{\text{gel}} = 2090 \) J/kg·°C (capacitat calorífica del gel) - \( L_f = 334,000 \) J/kg (calor latent de fusió de l'aigua) Substituint: \[ Q_{\text{gel}} = m_{\text{gel}} \cdot (2090 \cdot 18 + 334,000 + 4186 \cdot 5) \] \[ Q_{\text{gel}} = m_{\text{gel}} \cdot (37,620 + 334,000 + 20,930) \]Equilibri tèrmic:
\[ Q_{\text{beguda}} = Q_{\text{gel}} \] Resolent per \(m_{\text{gel}}\): \[ m_{\text{gel}} = \frac{27,628.8}{392,550} = 0.07 \, \text{kg} = 70 \, \text{g} \]Es calcula la calor específica de la barreja:
\[ c_{\text{mixt}} = x_{\text{aigua}} c_{\text{aigua}} + x_{\text{alcohol}} c_{\text{alcohol}} \] On: - \(x_{\text{aigua}} = 0.55\), \(x_{\text{alcohol}} = 0.45\) - \(c_{\text{aigua}} = 4186\,\text{J/kg·°C}\), \(c_{\text{alcohol}} = 2400\,\text{J/kg·°C}\) \[ c_{\text{mixt}} = (0.55)(4186) + (0.45)(2400) = 3402\,\text{J/kg·°C} \]Energia a dissipar per la beguda:
\[ Q_{\text{beguda}} = m_{\text{beguda}} \cdot c_{\text{mixt}} \cdot \Delta T \] \[ Q_{\text{beguda}} = 0.33 \cdot 3402 \cdot (25 - 5) = 22,452\,\text{J} \]Energia absorbida pel gel:
\[ Q_{\text{gel}} = m_{\text{gel}} \cdot c_{\text{gel}} \cdot \Delta T + m_{\text{gel}} \cdot L_f + m_{\text{gel}} \cdot c_{\text{aigua}} \cdot 5 \] Substituint: \[ Q_{\text{gel}} = m_{\text{gel}} \cdot (37,620 + 334,000 + 20,930) \]Equilibri tèrmic:
\[ Q_{\text{beguda}} = Q_{\text{gel}} \] Resolent per \(m_{\text{gel}}\): \[ m_{\text{gel}} = \frac{22,452}{392,550} = 0.057\,\text{kg} = 57\,\text{g} \]