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:
\[ Q_{\text{beguda}} = m_{\text{beguda}} \cdot c_{\text{mixt}} \cdot \Delta T \] \[ Q_{\text{beguda}} = 0.33 \cdot 3402 \cdot (25 - 5) = 20,330.7\,\text{J} \]Masses i densitat de la beguda (45% alcohol, 55% aigua):
\[ \rho_{\text{beguda}} = 0.55 \cdot \rho_{\text{aigua}} + 0.45 \cdot \rho_{\text{alcohol}} \] On: - \( \rho_{\text{aigua}} = 1000\,\text{kg/m}^3 \), \( \rho_{\text{alcohol}} = 789\,\text{kg/m}^3 \) \[ \rho_{\text{beguda}} = (0.55)(1000) + (0.45)(789) = 905.05\,\text{kg/m}^3 \] Per una beguda de 0.33 L: \[ m_{\text{beguda}} = \rho_{\text{beguda}} \cdot V = 905.05 \cdot 0.00033 = 0.2987\,\text{kg} = 298.7\,\text{g} \]Energia absorbida pel gel:
\[ Q_{\text{gel}} = m_{\text{gel}} \cdot (37,620 + 334,000 + 20,930) \] \[ Q_{\text{gel}} = m_{\text{gel}} \cdot 392,550 \]Equilibri tèrmic:
\[ Q_{\text{beguda}} = Q_{\text{gel}} \] Resolent per \(m_{\text{gel}}\): \[ m_{\text{gel}} = \frac{20,330.7}{392,550} = 0.0518\,\text{kg} = 52\,\text{g} \]